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\begin{document}

\begin{center}
\Large Graduate learning seminar on $\infty$-categories\\
\normalsize at The University of Illinois at Chicago\\[5pt]
Mondays 10-11 AM, SEO 1227\\
Fall 2017 semester\\[5pt]
\footnotesize This version typeset \today\\[5pt]
\hrule${}$\\[2pt]
\hrule
\end{center}

%\setcounter{tocdepth}{1}
\tableofcontents
\vspace{10pt}
\hrule

%\vfill

%\begin{center}
%\footnotesize\textit{love is not just a verb it's you looking in the mirror}
%\end{center}

\clearpage
\section{Simplicial sets}
\sectionsubtitle{\textit{J\=anis Lazovskis, 2017-09-11}}

Sources for this talk: \cite{groth} Section 1.1, \cite{goerss-jardine} Chapter 1, \cite{riehl} Sections 2 and 3.

\subsection{The category of simplicial sets}\label{sset}

First we introduce some categories.\\[5pt]
\label{not-delta}
\begin{tabular}{r l l}
$\Delta$: & Objects are $[n]=(0,1,\dots,n)$ \\ & \multicolumn{2}{l}{Morphisms are non-decreasing (equivalently order-preserving) maps $[n]\to[m]$}\\[5pt]
& Every morphism is a composition of: & \emph{coface maps} $s_i:[n]\to [n-1]$, hits $i$ twice \\
&& \emph{codegeneracy maps} $d_i:[n]\to [n+1]$, skips $i$ \\[5pt]
& For example: \\
& \multicolumn{2}{c}{
\begin{tikzpicture}[xscale=1.5,yscale=.7]
\foreach \x\y\namme in {0/1/1c, 0/2/1b, 0/3/1a, 1/0/2d, 1/1/2c, 1/2/2b, 1/3/2a}{
  \coordinate (\namme) at (\x,\y);
  \node[scale=.8] at (\x,\y) {$\bullet$};
}
\foreach \a\b in {1a/2a, 1b/2d, 1c/2d}{
  \draw[->,shorten >=5pt, shorten <=5pt] (\a)--(\b);
}
\begin{scope}[shift={(2.5,0)}]
\foreach \x\y\namme in {0/1/1c, 0/2/1b, 0/3/1a, 1/2/2b, 1/3/2a, 2/1/3c, 2/2/3b, 2/3/3a, 3/0/4d, 3/1/4c, 3/2/4b, 3/3/4a}{
  \coordinate (\namme) at (\x,\y);
  \node[scale=.8] at (\x,\y) {$\bullet$};
}
\foreach \a\b in {1a/2a, 1b/2b, 1c/2b, 2a/3a, 2b/3c, 3a/4a, 3b/4b, 3c/4d}{
  \draw[->,shorten >=5pt, shorten <=5pt] (\a)--(\b);
}
\node at (.5,3.5) {$s_1$};
\node at (1.5,3.5) {$d_1$};
\node at (2.5,3.5) {$d_2$};
\end{scope}
\begin{scope}[shift={(7,0)}]
\foreach \x\y\namme in {0/1/1c, 0/2/1b, 0/3/1a, 1/0/2d, 1/1/2c, 1/2/2b, 1/3/2a, 2/1/3c, 2/2/3b, 2/3/3a, 3/0/4d, 3/1/4c, 3/2/4b, 3/3/4a}{
  \coordinate (\namme) at (\x,\y);
  \node[scale=.8] at (\x,\y) {$\bullet$};
}
\foreach \a\b in {1a/2a, 1b/2c, 1c/2d, 2a/3a, 2b/3b, 2c/3c, 2d/3c, 3a/4a, 3b/4b, 3c/4d}{
  \draw[->,shorten >=5pt, shorten <=5pt] (\a)--(\b);
}
\node at (.5,3.5) {$d_1$};
\node at (1.5,3.5) {$s_2$};
\node at (2.5,3.5) {$d_2$};
\end{scope}
\node at (1.75,2) {is};
\node at (6.25,2) {or};
\end{tikzpicture}
}\\[10pt]
$[n]$: & Objects are numbers $0,1,\dots,n$ \\ & $|\Hom_{[n]}(a,b)| = 1$ iff $a\leqslant b$, else $\emptyset$\\[10pt]
$\sSet$: & $\Fun(\Delta^{op},\Set)$ \\[5pt]
& \multicolumn{2}{l}{An object (functor) may be described as $S = \{S_n=S([n])\}_{n\geqslant 0}$ with} \\
&& \emph{face maps} $S(s_i):S_n\to S_{n+1}$ \\
&& \emph{degeneracy maps} $S(d_i):S_n\to S_{n-1}$ \\[5pt]
& \multicolumn{2}{l}{Morphisms $f:S\to T$ are natural transformations} \\[10pt]
$\sObj$: & $\Fun(\Delta^{op},C)$ for $C$ any category
\end{tabular}

\begin{rem}
The final object in $\sSet$, denoted by $*$, is the functor that takes everything to the empty set.
\end{rem}

\subsection{Examples of simplicial sets}

Now we go through several examples of simplicial sets.

\begin{ex}
$\Delta^n = \Hom_{\Delta}(-,[n])$, called the \emph{standard $n$-simplex}. For $n=2$:
\[
\begin{tikzpicture}
\foreach \r\nam\lab in {90/a/1,210/b/0,330/c/2}{
  \draw[fill=black] ($(0,0)+(\r:1)$) circle (.05);
  \coordinate (\nam) at ($(0,0)+(\r:1)$);
  \node[scale=.8] at ($(0,0)+(\r:1.2)$) {\lab};  
}
%\fill[fill=gray!50] ($(0,0)+(90:.8)$)--($(0,0)+(210:.8)$)--($(0,0)+(330:.8)$);
\foreach \x\y in {a/b, b/c, c/a}{
  \draw[shorten >=4pt, shorten <=4pt] (\x)--(\y);
}
\node at (0,-1) {$[2]$};
\begin{scope}[shift={(-7,0)}]
\foreach \r\nam\lab in {90/1a/1,210/1b/0}{
  \draw[fill=black] ($(0,0)+(\r:1)$) circle (.05);
  \coordinate (\nam) at ($(0,0)+(\r:1)$);
  \node[scale=.8] at ($(0,0)+(\r:1.2)$) {\lab};  
}
\draw[shorten >=4pt, shorten <=4pt] (1a)--(1b);
\node at (0,-1) {$[1]$};
\end{scope}
\begin{scope}[shift={(-.5,.5)}]
\draw[->] (-5.5,1) to node[above=-2pt,scale=.8] {$0\mapsto0,\ 1\mapsto1$} (-1.5,1);
\draw[->] (-5.5,.5) to node[above=-2pt,scale=.8] {$0\mapsto1,\ 1\mapsto2$} (-1.5,.5);
\draw[->] (-5.5,0) to node[above=-2pt,scale=.8] {$0\mapsto0,\ 1\mapsto2$} (-1.5,0);
\draw[->] (-5.5,-.5) to node[above=-2pt,scale=.8] {$0,1\mapsto0$} (-1.5,-.5);
\draw[->] (-5.5,-1) to node[above=-2pt,scale=.8] {$0,1\mapsto1$} (-1.5,-1);
\draw[->] (-5.5,-1.5) to node[above=-2pt,scale=.8] {$0,1\mapsto2$} (-1.5,-1.5);
\end{scope}
\end{tikzpicture}
\]
Check contravariance: $[0]\to[1]$ with $0\mapsto 1$ should induce a map $\Hom_\Delta([1],[2])\to \Hom_\Delta([0],[2])$. It does, by pre-composition of $(\alpha:[1]\to[2])\in \Hom_\Delta([1],[2])$ by the same map $[0]\to[1]$.
\end{ex}
\label{not-deltan}


\begin{ex}
$\Sing(X)_n = \Hom_\Top(\Delta^n_{top},X)$, for $X$ a topological space. Contravariance works as above.
\end{ex}

\begin{rem}\index{adjunction}
There is an adjunction between the categories $\Top$ and $\sSet$, given by geometric realization $|-|$ in one direction and $\Sing(-)$ in the other. This allows for a more visual representation of simplicial sets.
\end{rem}

\begin{ex}\index{nerve}\label{not-nerve}
$N(C)_n = \Fun([n],C)$, for $C$ any category, called the \emph{nerve} of $C$. Note that
\begin{center}
\begin{tabular}{r c l}
$N(C)_0$ & $=$ & objects of $C$ \\
$N(C)_1$ & $=$ & morphisms of $C$ \\
$N(C)_2$ & $=$ & pairs of composable arrows of $C$\\
& $\vdots$ \\
$N(C)_n$ & $=$ & strings of $n$ composable arrows of $C$
\end{tabular}
\end{center}
For example, the two maps $d_1,d_0:[0]\to [1]$, given by $0\mapsto 0$ and $0\mapsto 1$, respectively, induce natural transformations from $F_1:[1]\to C$ to $F_0:[0]\to C$, which we may call the \emph{domain} and \emph{range}.
\[
\begin{tikzpicture}
\node[anchor=east] (0) at (0,2) {$[0]$};
\node[anchor=east] (01) at (0,1) {$0$};
\node[anchor=east] (02) at (0,0) {$0$};
\node[anchor=west] (1) at (1,2) {$[1]$};
\node[anchor=west] (11) at (1,1) {$0$};
\node[anchor=west] (12) at (1,0) {$1$};
\node[anchor=east] at (-.5,1) {$d_1:$};
\node[anchor=east] at (-.5,0) {$d_0:$};
\node[anchor=east] at (-1.5,1) {``domain"};
\node[anchor=east] at (-1.5,0) {``range"};
\draw[->] (0)--(1);
\draw[|->] (01)--(11);
\draw[|->] (02)--(12);
\node at (3,1) {$\rightsquigarrow$};
\begin{scope}[shift={(5.5,0)}]
\node[anchor=east] (f1) at (2,2) {$(F_1:[1]\to C)$};
\node[anchor=west] (f0) at (3,2) {$(F_0:[0]\to C)$};
\node[anchor=east] (a1) at (2,1) {$(\alpha\in \Hom_C(A,B))$};
\node[anchor=east] (b1) at (2,0) {$(\alpha\in \Hom_C(A,B))$};
\node[anchor=west] (a2) at (3,1) {$A$};
\node[anchor=west] (b2) at (3,0) {$B$};

\draw[->] (f1)--(f0); 
\draw[|->] (a1)--(a2);
\draw[|->] (b1)--(b2);
\end{scope}
\end{tikzpicture}
\]
Note also that the face map $N(s_i):\Fun([n],C)\to \Fun([n+1],C)$ inserts the identity arrow at the $i$th spot, and the degeneracy map $N(d_i):\Fun([n],C)\to \Fun([n+1],C)$ composes the $i$th and $(i+1)$th arrows.
\end{ex}

\begin{ex}
$N([n]) = \Delta^n$, the standard $n$-simplex.
\end{ex}

\begin{ex}\index{horn}\label{not-horn}
$\Lambda^n_i$, the \emph{$i$th $n$-horn}. Heuristically, it is generated by all elements of $\Delta^n$ except the $i$th face. Formally,
\[
\Lambda_i^n = \displaystyle\bigcup_{\gfrac{\alpha\in \Delta^n}{j\neq i}} d_j\circ s_j \circ \alpha.
\]
There is a natural inclusion map (that is, inclusion natural transformation) $i:\Lambda^n_i\hookrightarrow \Delta^n$ for all $0\leq i\leq n$.
\end{ex}

\subsection{Fibrations, Yoneda, and more examples}

Before we talk about more examples, we need to introduce a new definition.

\begin{defn}\index{fibration}
Let $f:S\to T$ be a morphism of simplicial complexes (that is, a natural transformation). Then $f$ is a \emph{fibration} if for every pair of morphisms $\Lambda^n_i\to S$ and $\Delta^n\to T$ such that the diagram on the left commutes, there exists a map $\Delta^n\to S$, so that the diagram on the right still commutes.
\[
\begin{tikzpicture}[yscale=1.2,xscale=1.3]
\node (d) at (0,0) {$\Delta^n$};
\node (h) at (0,1) {$\Lambda^n_i$};
\node (t) at (1,0) {$T$};
\node (s) at (1,1) {$S$};
\draw[{Hooks[right]}->] (h) to node[left] {$i$} (d);
\draw[->] (s) to node[right] {$f$} (t);
\draw[->] (h)--(s);
\draw[->] (d)--(t);
\end{tikzpicture}
\hspace{2cm}
\begin{tikzpicture}[yscale=1.2,xscale=1.3]
\node (d) at (0,0) {$\Delta^n$};
\node (h) at (0,1) {$\Lambda^n_i$};
\node (t) at (1,0) {$T$};
\node (s) at (1,1) {$S$};
\draw[{Hooks[right]}->] (h) to node[left] {$i$} (d);
\draw[->] (s) to node[right] {$f$} (t);
\draw[->] (h)--(s);
\draw[->] (d)--(t);
\draw[->,dashed] (d)--(s);
\end{tikzpicture}
\]
\end{defn}

\begin{defn}\index{fibrant!simplicial set}\index{simplicial set!fibrant}\index{Kan complex}\index{simplicial set!fibrant}
An object $S\in \sSet$ is \emph{fibrant}, or a \emph{Kan complex}, if either of the equivalent conditions is satsifed:
\begin{itemize}
\item The canonical map $S\to *$ is a fibration.
\item Every map $\Lambda^n_i\to S$ may be extended to a map $\Lambda^n_i\hookrightarrow \Delta^n\to S$.
\end{itemize}
\end{defn}

\begin{ex}
$\Sing(X)$ is a Kan complex, for $X$ a topological space.
\end{ex}

\begin{ex}
$\Pi_1(X)$, the fundamental groupoid of a topological space $X$, is a Kan complex. Recall that objects are points in $X$ and morphism from $x$ to $y$ is a homotopy classes of continuous maps $\gamma:[0,1]\to X$ with $\gamma(0)=x$ and $\gamma(1)=y$. Hence the relation to the fundamental group is
\[
\pi_1(X,x) = \textup{Aut}_{\Pi_1(X)}(x).
\]
It is a simplicial set because it is the nerve of the category of points of $X$ and paths between points. {\color{red}Not sure why this is a Kan complex.}%It is a Kan complex because ``edges of simplices may be homotoped to each other".
\end{ex}

We finish with a useful statement.

\begin{lem}\index{Yoneda lemma}\index{lemma!Yoneda}{\sc[Yoneda]} \\
Let $S\in \sSet$. Morphisms $\Delta^n\to S$ in $\sSet$ (that is, natural transformations) correspond bijectively to elements of $S_n$. Moreover, the correspondence is natural in both directions.
\end{lem}

In other words, $\Hom_\sSet(\Delta^n,S)\cong S_n$. This statement makes it easier to describe morphisms between simplicial sets (that is, natural transformations).\\

\hrule
\section{$\infty$-categories via Grothendieck}
\sectionsubtitle{\textit{Micah Darrell, 2017-09-18}}

\subsection{Motivation for higher categories}

In ``Pursuing Stacks," Grothendieck suggested there should a notion of ``higher groupoids," with an ``$n$-groupoid" modeling an ``$n$-type."\\

\noindent
\underline{1-groupoid}: A category whose morphisms are all invertible. For example, a group is a groupoid with a single object (the set of elements of the group), and morphisms are multipliation by the group elements. We also have the \emph{fundamental groupoid} $\Pi(X)$ for any topological space $X$, and the \emph{classifying space} $BG$ for any group $G$.\\
\index{fundamental groupoid}\index{groupoid}\index{classifying space}


\noindent
\underline{2-category}: Contains objects, morphisms, and ``morphisms between morphisms." For example, given $f:A\to B$, $g:B\to C$, and $h:C\to D$ all morphisms, we have two notions of a 2-category:\\
\indent \emph{srict} 2-category: if $h\circ (g\circ f) = (h\circ g)\circ f$ \\
\indent \emph{lax} 2-category: if $h\circ (g\circ f) \cong (h\circ g)\circ f$

\begin{thm}{\sc[Mac Lane's coherence theorem]}\index{Mac Lane's coherence theorem}\index{theorem!Mac Lane's coherence}\\
Stict and lax 2-categories are the same, up to 2-categorical equivalence.
\end{thm}

\index{delooping}
Issues arise in 3-categories. Lurie in \cite{lurie-htt} mentions that ``no strict 3-groupoid can model $S^2$," where ``to model" means to go functorially from an $n$-groupoid to a space. If functoriality were possible, it should be that the delooping of a groupoid should be the delooping of the space. Recall:
\begin{itemize}
\item The \emph{delooping} of an $n$-groupoid $G$ is an $(n+1)$-groupoid $H$ with one object $X$, such that $\Hom_H(X,X)=G$.
\item The \emph{delooping} of a space $X$ is a space $F$ for which $X \cong \Omega F$.
\end{itemize}
Suppose we have an $n$-groupoid $G$, whose delooping is an $(n+1)$-groupoid $H$, a strict $(n+1)$-category. This gives a monoidal structure to $G$ (meaning there is a functor $G\times G\to G$ satisfying some properties), call it $\nu:G\times G\to G$. If we can deloop $H$, then we get $\mu:H\times H\to H$, and this restricts to a new monoidal structure on $G$, call it $\mu_G:G\times G\to G$.The Eckmann--Hilton argument gives that $G$ and $H$ are commutative monoids, so $\mu_G$ and $|nu$ are the same, and $G$ is a commutative monoid.\\

The above shows that an $n$-groupoid can be delooped infinitely many times, but a 2-sphere can only be delooped twice, confirming Lurie's statement.

\begin{rem}
To make things simpler (although losing some properties of higher morphisms), we use $(\infty,n)$-categories to specify non-strict $n$-categories. We will be interested in $(\infty,1)$-categories, which meand we have morphisms ar every level, and for $k>1$ they are invertible.
\end{rem}

\subsection{Defining $\infty$-categories}

Grothendieck's homotopy hypothesis says that $(\infty,0)$-categories are spaces. Informally:

\begin{defn}
An $(\infty,1)$-category is a category enriched in spaces.
\end{defn}

\index{enriched category}\index{category!enriched}
\begin{defn}
A category $\mathcal C$ is \emph{enriched} in a category $\mathcal D$ if\\
\indent1.\ $\mathcal D$ has a monoidal structure, \\
\indent2.\ $\Hom_{\mathcal C}(X,Y)\in \mathcal D$, and \\
\indent3.\ composition in $\mathcal C$ is given by the monoidal structure on $\mathcal D$.
\end{defn}

This way of defining $\infty$-categories models some properties of spaces, but it is difficult to use. We can also relate an $(\infty,1)$-category $\mathcal C$ to a 1-category $\Ho(\mathcal C)$, called the \emph{homotopy category} of $\mathcal C$, which has the same objects as $\mathcal C$ and $\Hom_{\Ho(\mathcal C)}(X,Y) = \Pi_0\Map_{\mathcal C}(X,Y)$.\\
\index{homotopy category}\index{category!homotopy}\label{not-homcat}

Recall from last time we had a space $X$ and a Kan complex $\Sing(X)$, which meant that any map $\Lambda^n_i\to X$ from the $i$th $n$-horn can be lifted to a map $\Delta^n\to X$ from the $n$-simplex, for any $n$ and all $0<i<n$, called the \emph{inner horn condition}. For any 1-category $\mathcal C$, we also had a simplicial set $N(\mathcal C)$, the nerve, for which the lifting was unique. For example, if $n=2$ and $i=1$, compositions are unique, so the lifting is unique.
\[
\begin{tikzpicture}
\foreach \r\nam\lab in {90/a/B,210/b/A,330/c/C}{
  \draw[fill=black] ($(0,0)+(\r:1)$) circle (.05);
  \coordinate (\nam) at ($(0,0)+(\r:1)$);
  \node[scale=.8] at ($(0,0)+(\r:1.3)$) {$\lab$};  
}
\foreach \x\y\lab\nod in {b/a/f/{auto}, b/c/{g\circ f}/{below}, a/c/g/{auto}}{
  \draw[->,shorten >=4pt, shorten <=4pt] (\x) to node[\nod] {$\lab$} (\y);
}
\end{tikzpicture}
\]

\begin{defn} (after Quillen)
An $\infty$-category is a simplicial set $X$ with the inner horn condition.
\end{defn}

Recall the \emph{inner horn} is for $0<i<n$, and the \emph{outer horn} is for $i=0,n$. Moreover if the inner horn condition gives a unique lifting, then the $\infty$-category is modeled by the nerve of some other category. We will denote the objects of an $\infty$-category $\mathcal C$ by $\mathcal C_0$, the morphisms by $\mathcal C_1$, and so on.\\

Note that horn liftings need not always be unique. For example, when $n=2$ and $i=1$, we can have
\[
\begin{tikzpicture}[baseline=-.1cm]
\foreach \r\nam\lab in {90/a/B,210/b/A,330/c/C}{
  \draw[fill=black] ($(0,0)+(\r:1)$) circle (.05);
  \coordinate (\nam) at ($(0,0)+(\r:1)$);
  \node[scale=.8] at ($(0,0)+(\r:1.3)$) {$\lab$};  
}
\foreach \x\y\lab\nod in {b/a/f/{auto}, a/c/g/{auto}}{
  \draw[->,shorten >=4pt, shorten <=4pt] (\x) to node[\nod] {$\lab$} (\y);
}
\end{tikzpicture}
\hspace{1cm}
\text{filling to}
\hspace{1cm}
\begin{tikzpicture}[baseline=-.1cm]
\foreach \r\nam\lab in {90/a/B,210/b/A,330/c/C}{
  \draw[fill=black] ($(0,0)+(\r:1)$) circle (.05);
  \coordinate (\nam) at ($(0,0)+(\r:1)$);
  \node[scale=.8] at ($(0,0)+(\r:1.3)$) {$\lab$};  
}
\foreach \x\y\lab\nod in {b/a/f/{auto}, b/c/{h}/{below}, a/c/g/{auto}}{
  \draw[->,shorten >=4pt, shorten <=4pt] (\x) to node[\nod] {$\lab$} (\y);
}
\end{tikzpicture}
\hspace{1cm}
\text{or}
\hspace{1cm}
\begin{tikzpicture}[baseline=-.1cm]
\foreach \r\nam\lab in {90/a/B,210/b/A,330/c/C}{
  \draw[fill=black] ($(0,0)+(\r:1)$) circle (.05);
  \coordinate (\nam) at ($(0,0)+(\r:1)$);
  \node[scale=.8] at ($(0,0)+(\r:1.3)$) {$\lab$};  
}
\foreach \x\y\lab\nod in {b/a/f/{auto}, b/c/{k}/{below}, a/c/g/{auto}}{
  \draw[->,shorten >=4pt, shorten <=4pt] (\x) to node[\nod] {$\lab$} (\y);
}
\end{tikzpicture}
\ .
\]

\begin{defn}
Two 1-simplices (arrows) $f,g:X\to Y$ are \emph{homotopy equivalent} if there exists a 2-simplex 
\[
\begin{tikzpicture}
\foreach \r\nam\lab in {90/a/X,210/b/X,330/c/Y}{
  \draw[fill=black] ($(0,0)+(\r:1)$) circle (.05);
  \coordinate (\nam) at ($(0,0)+(\r:1)$);
  \node[scale=.8] at ($(0,0)+(\r:1.3)$) {$\lab$};  
}
\foreach \x\y\lab\nod in {b/a/{\id_X}/{auto}, b/c/{g}/{below}, a/c/f/{auto}}{
  \draw[->,shorten >=4pt, shorten <=4pt] (\x) to node[\nod] {$\lab$} (\y);
}
\end{tikzpicture}
\ .
\]
\end{defn}

\hrule
\section{Examples of $\infty$-categories}
\sectionsubtitle{\textit{Greg Taylor, 2017-09-26}}

Sources for this talk: \cite{freed}, \cite{lurie-otcotft}, \cite{camarena}, \cite{gaitsgory-rozenblyum}, \cite{toen}, nLab article ``Fundamental $\infty$-groupoid".

\subsection{The fundamental $\infty$-groupoid}

\index{fundamental groupoid}
Let $X$ be a topological space. Recall the functor $\Sing(X)$ that takes $[k]\in \Delta$ to $\Hom_{\Top}(|\Delta^k|,X)$. This is the simplicial set description of the \emph{fundamental $\infty$-groupoid} $\Pi_\infty(X)$. Note the \emph{fundamental groupoid} $\Pi_1(X)$ is the collection of 1-morphisms of $\Pi_\infty(X)$.

\begin{rem}\index{homotopy hypothesis}\index{hypothesis!homotopy}
There is an adjunction $|\ \cdot\ |:\infty\textup{-groupoid} \leftrightarrows \Top:\Pi_\infty(\ \cdot\ )$. This is the technical interpretation of the homotopy hypothesis.
\end{rem}

\subsection{Quasi-coherent sheaves}

Now let $X$ be a quasi-compact quasi-separated scheme. Without loss of generality, let $X$ be a variety over an algebraically closed field $k$.

\begin{defn}\index{quasi-coherent sheaf}\index{sheaf!quasi-coherent}\label{not-qcs}
A \emph{quasi-coherent} sheaf $\mathcal F$ on $X$ is a sheaf of $\mathcal O_X$-modules for which $X$ has an affine cover $\{\Spec(A_i)\}_i$ such that $\mathcal F|_{\Spec(A_i)} = \widetilde {M_i}$. Let $QC(X)$ be the category of quasi-coherent sheaves and sheaf morphisms on $X$.
\end{defn}

Recall the notation of sheaves:
\[
\widetilde{M_i}(D(f)) = (M_i)_f,
\hspace{1cm}
(\widetilde{M_i})_p = (M_i)_p,
\hspace{1cm}
D(f) = \{p\in \Spec(A_i)\ :\ f\not\in p\}\subseteq \Spec(A_i).
\]

Note that $QC(X)$ is symmetric monoidal, where the usual tensor product of sheaves gives it a symmetric monoidal structure. 

\begin{rem}\index{derived category}\index{category!derived}
Viewing $QC(X)$ as a subcategory of chain complexes, we get that $QC(X)$ is an $\infty$-category, call it $QC^\infty(X)$. Moreover, the two are related therough the homotopy category and derived categories. That is,
\[
\Ho(QC^\infty(X)) = D(QC(X)).
\]
Recall that the derived category $D(C)$ is constructed by first taking the category of complexes of $C$, then identifying chain homotopic morphisms, and finally localizing along quasi-isomorphisms (that is, quotienting by the quasi-isomorphism equivalence relation).
\end{rem}

However, we cannot just ``glue" derived categories to go from local to global properties. This does work in the $\infty$ setting. That is, if $\{\Spec(A_i)\}_i$ covers $X$, then
\[
\Ho(QC^\infty(X)) = \Ho\left(\ ``\!\!\!\!\lim_{\Spec(A_i)}\!\!\!\!\!"\ \ QC^\infty(\Spec(A_i))\right),
\]
where the limit has not yet been defined in the $\infty$ setting.

\subsection{TQFTs}

Consider the category $\Cob(d\in \Z_{>0})$ of closed, oriented, compact $(d-1)$-manifolds and \emph{cobordisms} between them.

\begin{defn}\index{cobordism}\index{boridsm}
Let $M,N$ be objects of $\Cob(d)$. A \emph{cobordism} (or \emph{bordism}) between $M$ and $N$ is a $d$-manifold $B$ for which $\partial B= M\sqcup N$.
\end{defn}

\begin{defn}
A \emph{topological quantum field theory} (or \emph{TQFT}) is a functor $Z:\Cob(d)\to \Vect(\C)$ with
\begin{itemize}
\item $Z(M\sqcup N) = Z(M)\otimes Z(N)$, and 
\item $Z(\emptyset) = \C$.
\end{itemize}
\end{defn}

The above classical definition is due to Atiyah. Note that for any closed, compact $d$-manifold $M$ without boundary, $Z(M:\emptyset\to\emptyset) = (\C\to \C)$ is just a $\C$-endomorphism.

\begin{thm}{\sc[Cobordism Hyopthesis, Baez--Dolan]}\index{cobordism hyopthesis}\index{hypothesis!cobordism} \\
Informally speaking, a TQFT is determined by its value on a point.
\end{thm}

To prove this, Ayala, Francis, and Lurie have employed the extended category $\Cob(d)^{fr}_{ext}$ of framed cobordisms, which involves a choice of basis of the vector space in the target category. This is an $\infty$-category, with objects points (0-dimensional manifolds), 1-morphisms cobordisms between objects (1-dimensional manifolds), and $d$-morphsism cobordisms of $(d-1)$-manifolds ($d$-dimensional manifolds).\\

\hrule
\section{Limits and colimits}
\sectionsubtitle{\textit{Joel Stapleton, 2017-10-02}}

Sources for this talk: \cite{groth}, Sections 2.2 to 2.5.

\subsection{Colimits in 1-categories}

We wil only talk about final objects and colimits, everything can be dualized for initial objects and limits. Let $\mathcal C$ be a category.

\begin{defn}\index{final object}
An object $X$ of $\mathcal C$ is \emph{final} in $\mathcal C$ if $\Hom_{\mathcal C}(Y,X)=*$ for all objects $Y$ in $\mathcal C$.
\end{defn}

\label{not-slice}
To extend colimits to higher categories, we define the category $\mathcal C_{p/}$, for every functor $p:\mathcal D\to \mathcal C$ of categories. This is called the \emph{slice category}, or the \emph{cocone of diagrams over $\mathcal C$}, and it may be viewed as a subcategory of $\Fun(\mathcal D,\mathcal C)$. For example, suppose we have a diagram
\index{cocone}
\[
\begin{tikzpicture}
\fill[fill=black] (0,0) circle (.05);
\fill[fill=black] (1,.75) circle (.05);
\fill[fill=black] (1,-.75) circle (.05);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0)--(1,.75);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0)--(1,-.75);
\end{tikzpicture}
\]
in $\mathcal D$. Apply the functor $p$ to this diagram to get a similar diagram in $\mathcal C$, whose cocone
\[
\begin{tikzpicture}
\fill[fill=black] (0,0) circle (.05);
\fill[fill=black] (1,.75) circle (.05);
\fill[fill=black] (1,-.75) circle (.05);
\fill[fill=black] (2,0) circle (.05);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0)--(1,.75);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0)--(1,-.75);
\draw[->,shorten >=4pt, shorten <=4pt,densely dashed] (0,0)--(2,0);
\draw[->,shorten >=4pt, shorten <=4pt,densely dashed] (1,.75)--(2,0);
\draw[->,shorten >=4pt, shorten <=4pt,densely dashed] (1,-.75)--(2,0);
\end{tikzpicture}
\]
is another object of $\mathcal C$ and morphisms into that object, so that the diagram commutes.

\begin{defn}\index{colimit}\index{limit}
A \emph{colimit} of $p$ is an initial object in $\mathcal C_{p/}$.
\end{defn}

Actually, it is not quite an initial object, but the composition of the initial object with the constant functor that gives the target object of a morphism.

\subsection{The join construction in 1-categories}

Let $A,B$ be categories.

\begin{defn}\index{join}\label{not-join}
The \emph{join} $A\star B$ of $A$ and $B$ is another category, for which
\[
\begin{array}{r c l}
\obj(A\star B) & = & \obj(A) \coprod \obj(B), \\[10pt]
\hom(X,Y) & = & \begin{cases}
\hom_A(X,Y) & \text{\ if\ }X,Y\in \obj(A), \\
\hom_B(X,Y) & \text{\ if\ }X,Y\in \obj(B), \\
* & \text{\ if\ } X\in \obj(A), Y\in \obj(B), \\
\emptyset & \text{else.}
\end{cases}
\end{array}
\]
\end{defn}

The idea is to think of this as connecting two categories together with morphisms.

\begin{ex}
For $\mathbf{1}$ the terminal object in the category of small categories and $\mathcal C$ any other category, the join $\mathcal C\star \mathbf{1}$ is simply the cocone of $\mathcal C$, and $\mathbf{1}\star \mathcal C$ is the cone of $\mathcal C$.
\end{ex}

The join is not symmetric in general, but it is symmetric in the geometric interpretation. 

\begin{ex}
Consider two simplices $[i]$ and $[j]$ as categories (as defined in Section \ref{sset}). Their join is
\[
[i]\star[j] = [i+j+1].
\]
Let us check this for $i=j=0$. We begin with two 0-simplices
\[
\begin{tikzpicture}
\fill[fill=black] (0,0) circle (.05);
\fill[fill=black] (1,0) circle (.05);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0).. controls +(50:.8) and +(130:.8)..(0,0);
\draw[->,shorten >=4pt, shorten <=4pt] (1,0).. controls +(50:.8) and +(130:.8)..(1,0);
\end{tikzpicture}
\]
and add on a single element for a morphism between them, giving
\[
\begin{tikzpicture}
\fill[fill=black] (0,0) circle (.05);
\fill[fill=black] (1,0) circle (.05);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0).. controls +(50:.8) and +(130:.8)..(0,0);
\draw[->,shorten >=4pt, shorten <=4pt] (1,0).. controls +(50:.8) and +(130:.8)..(1,0);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0)--(1,0);
\end{tikzpicture}
\]
which is indeed the 1-simplex $[1]$.
\end{ex}

We are trying to get a universal characterization of the slice category. The following statement gives us that.

\begin{prop}
$\Fun(A,\mathcal C_{/p})\cong \Fun_p(A\star \mathcal D, \mathcal C)$.
\end{prop}

The category on the right may be viewed as all the morphisms that makes diagram of the type
\[
\begin{tikzpicture}
\node (d) at (0,0) {$\mathcal D$};
\node (j) at (-1,-1) {$A\star \mathcal D$};
\node (c) at (1,-1) {$\mathcal C$};
\draw[->] (d) to node[auto,swap] {$i$} (j);
\draw[->] (d) to node[auto] {$p$} (c);
\draw[->,densely dashed] (j)--(c);
\end{tikzpicture}
\]
commute.

\subsection{The join construction in simplicial sets}

Now we translate joins into the laguage of simplicial sets, which should account for all the higher dimensional objects. Let $K,L$ be simplicial sets, or elements of $\obj(\sSet)$.

\begin{defn}\index{join}
The \emph{join} $K\star L$ of $K$ and $L$ is another simplicial set, for which
\[
(K\star L)_n = K_n\cup L_n\bigcup_{i+j+1=n}K_i\times L_j.
\]
\end{defn}

\begin{ex}
Let us check this for small $n$. When $n=0,1$, we have
\[
(K\star L)_0 = K0\cup L_0,
\hspace{1cm}
(K\star L)_1 = K_1\cup L_1 \cup (K_0\times L_0).
\]
\end{ex}

To take this to the subcategory of $\infty$-categories, we need to make sure it is closed and preserves equivalences there. We cite a proposition from \cite{groth}.

\begin{prop}${}$\\
1. If $\mathcal C,\mathcal D$ are $\infty$-categories, then $\mathcal C\star \mathcal D$ is an $\infty$-category. \\
2. If $f:\mathcal C\to \mathcal D$ and $g:\mathcal C'\to \mathcal D'$ are equivalences of $\infty$-categories, then $f\star g$ is also an equivalence of $\infty$-categories.
\end{prop}

The map $f\star g:\mathcal C\star \mathcal C'\to \mathcal D\star \mathcal D'$ is defined in the natural way. An \emph{equivalence} of $\infty$-categories may be thought of a particular morphism of simplicial sets (which is a natural transformation between two functors out of $\Delta^{op}$), although defining it precisely requires the notion of an \emph{$\infty$-morphism}, which we do not attempt to describe here.

\begin{prop}
$N(A\star B) \cong N(A)\star N(B)$.
\end{prop}

\begin{defn}
Let $p:L\to \mathcal C$ be a map of simplicial sets, with $\mathcal C$ an $\infty$-category. There is an $\infty$-category $\mathcal C_{/p}$ characterized by the following universal property: For any $K\in \obj(\sSet)$, 
\[
\hom_{\sSet}(K,\mathcal C) \cong \hom_{\sSet\ L/}(L\to K\star L,L\xrightarrow{\ p\ }\mathcal C).
\]
\end{defn}

For example, $(\mathcal C_{/p})_n = \hom_{\sSet}([n],\mathcal C_{/p}) = \hom_{\sSet\ L/}(L\to [n]\star L,L\xrightarrow{\ p\ }\mathcal C)$.

\begin{lem}
If $p:A\to B$ is a morphsim of categories, then $N(B_{/p}) \cong N(B)_{/p}$.
\end{lem}

\begin{proof} (Sketch)
Observe that
\begin{align*}
N(B_{/p})_n & \cong \hom_{\sSet}([n],N(B_{/p})) \\
& \cong \hom_{\Cat}([n],B_{/p}) & \text{(by full faithfulness of hom-sets)} \\
& \cong \hom_{\Cat\ A/}(A\to [n]\star A,A\to B) & \text{(by characterization of join)} \\
& \cong \hom_{\sSet\ N(A)/}(N(A)\to [n]\star N(A),N(A)\xrightarrow{\ N(p)\ }N(B)) & \text{(take the nerve)} \\
& \cong \hom_{\sSet}([n],N(B)_{/N(p)}).
\end{align*}
\end{proof}
 We now need to describe what final objects are in $\infty$-categories.

\subsection{Final objects in $\infty$-categories}

\begin{defn}\index{final object}
An object $X$ in an $\infty$-category $\mathcal C$ is \emph{final} if $\mathcal C_{/X}\to \mathcal C$ is an acyclic fibration of simplicial sets.
\end{defn}

An immediate corollary of this definition is the following:

\begin{prop}
Let $\mathcal C$ be an $\infty$-category and $\mathcal D\subseteq \mathcal C$ be the full subcategory of final objects. Then $\mathcal D$ is either empty or a contractible Kan complex.
\end{prop}

\begin{defn}\index{colimit}\index{limit}
Let $K$ be a simplicial set and $\mathcal C$ an $\infty$-category. A \emph{colimit} of a morphism $p:K\to \mathcal C$ is an initial object of $\mathcal C_{p/}$.
\end{defn}

\hrule
\section{Model categories}
\sectionsubtitle{\textit{Harry Smith, 2017-10-09}}

Sources for this talk: \cite{dwyer-spalinski}\\

\noindent
Newt week we will see an equivalence between presentable $\infty$-categories and combinatorial model categories, this week will be setting up all the necessary definitions.

\subsection{The model category axioms}

Before we can give the axioms for a model category, we need some preliminary definitions.

\begin{defn}\index{retract}\index{left lifting property}\index{right lifting property}\index{lifting property}
A morphism $f\in \Hom(A,B)$ is a \emph{retract} of $g\in \Hom(C,D)$ if the diagram
\[
\begin{tikzpicture}[scale=1.3]
\node (a1) at (0,1) {$A$};
\node (b1) at (0,0) {$B$};
\node (c) at (1,1) {$C$};
\node (d) at (1,0) {$D$};
\node (a2) at (2,1) {$A$};
\node (b2) at (2,0) {$B$};
\foreach \x\y\n in {a1/b1/f, a2/b2/f, c/d/g, a1/c/{}, c/a2/{}, b1/d/{}, d/b2/{}}{
  \draw[->] (\x) to node[left] {$\n$} (\y);
}
\draw[->] (a1) to [bend left=20] node[above] {id} (a2);
\draw[->] (b1) to [bend right=20] node[below] {id} (b2);
\end{tikzpicture}
\]
commutes. Further, $f$ has the \emph{left lifting property} (LLP) with respect to $g$, and $g$ has the \emph{right lifting property} (RLP) with respect to $f$ if for every commuting diagram without the dashed arrow, the dashed arrow exists, keeping commutativity.
\[
\begin{tikzpicture}[yscale=1.2,xscale=1.3]
\node (d) at (0,0) {$B$};
\node (h) at (0,1) {$A$};
\node (t) at (1,0) {$D$};
\node (s) at (1,1) {$C$};
\draw[->] (h) to node[left] {$f$} (d);
\draw[->] (s) to node[right] {$g$} (t);
\draw[->] (h)--(s);
\draw[->] (d)--(t);
\draw[->,dashed] (d)--(s);
\end{tikzpicture}
\]
\end{defn}

Now we have enough to define model categories.

\begin{defn}\label{not-modelcat}\index{model category}\index{category!model}\index{acyclic fibration}\index{acyclic cofibration}\index{fibration}\index{cofibration}\index{weak equivalence}
A \emph{model category} is a category $\mathcal C$ with three classes $F$ (denoting \emph{fibrations} $\twoheadrightarrow$), $C$ (denoting \emph{cofibrations} $\hookrightarrow$), and $W$ (denoting \emph{weak equivalences} $\xrightarrow\sim$) of distinguished morphisms, satisfying the conditions below. Elements of $F\cap W$ are called \emph{acyclic fibrations} $\stackrel{\sim}{\twoheadrightarrow}$ and elements of $C\cap W$ are called \emph{acyclic cofibrations} $\stackrel{\sim}{\hookrightarrow}$.
\begin{enumerate}
\item $\mathcal C$ is complete and cocomplete (limits and colimits exist).
\item Weak equivalences satisfy the two-out-of-three rule. That is, if any of two of the thee arrows in the commuting diagram
\[
\begin{tikzpicture}
\foreach \r\nam\lab in {90/a/X,210/b/X,330/c/Y}{
  \draw[fill=black] ($(0,0)+(\r:1)$) circle (.05);
  \coordinate (\nam) at ($(0,0)+(\r:1)$);
}
\foreach \x\y\lab\nod in {b/a/{\id_X}/{auto}, b/c/{g}/{below}, a/c/f/{auto}}{
  \draw[->,shorten >=4pt, shorten <=4pt] (\x) to (\y);
}
\end{tikzpicture}
\]
of objects and morphisms in $\mathcal C$ are weak equivalences, then the third one is as well.
\item Whenever $f$ is a retract of $g$, if $g$ is in $F,C,W$, then $f$ is in $F,C,W$, respectively.
\item A cofibration has the LLP with respect to any acyclic fibration. A fibration has the RLP with respect to any acyclic cofibration.
\item Every morphism $\begin{tikzpicture}[baseline=-.1cm]
\fill[fill=black] (0,0) circle (.05);
\fill[fill=black] (1,0) circle (.05);
\draw[->,shorten >=5pt, shorten <=5pt] (0,0)--(1,0);
\end{tikzpicture}$ can be factored as an acyclic cofibration followed by a fibration, or a cofibration followed by an acyclic fibration. That is, we have the following commutative diagram.
\[
\begin{tikzpicture}
\fill[fill=black] (0,0) circle (.05);
\fill[fill=black] (2,0) circle (.05);
\fill[fill=black] (1,1) circle (.05);
\fill[fill=black] (1,-1) circle (.05);
\draw[->,shorten >=5pt, shorten <=5pt] (0,0)--(2,0);
\draw[{Hooks[right]}->,shorten >=5pt, shorten <=5pt] (0,0) to node[rotate=45,above=-2pt,pos=.5] {$\sim$} (1,1);
\draw[->>,shorten >=5pt, shorten <=5pt] (1,1)--(2,0);
\draw[{Hooks[right]}->,shorten >=5pt, shorten <=5pt] (0,0)--(1,-1);
\draw[->>,shorten >=5pt, shorten <=5pt] (1,-1) to node[rotate=45,above=-2pt,pos=.4] {$\sim$} (2,0);
\end{tikzpicture}
\]
\end{enumerate} 
\end{defn}

\subsection{Fibrations and cofibrations}

The initial object of a model category $\mathcal C$ is denoted $\emptyset$ and the final object is $*$.

\begin{defn}\label{not-fibcofib}\index{fibrant!object}\index{cofibrant!object}\index{bifibrant!object}\index{fibrant!replacement}\index{cofibrant!replacement}\index{bifibrant!replacement}\index{object!fibrant}\index{object!cofibrant}\index{object!bifibrant}\index{replacement!fibrant}\index{replacement!cofibrant}\index{replacement!bifibrant}
An object $A$ of a model category $\mathcal C$ is \emph{fibrant} if $A\twoheadrightarrow *$ and \emph{cofibrant} if $\emptyset \hookrightarrow A$. That is, if the unique maps into $*$ and out of $\emptyset$ are fibrations and cofibrations, respectively. An object that is both fibrant and cofibrant is called \emph{bifibrant}. The unique objects $A_F$ and $A_C$ that make the middle three levels of the diagram
\[
\begin{tikzpicture}[xscale=1.2,yscale=1.5]
\node (i) at (0,0) {$\emptyset$};
\node (a) at (2,0) {$A$};
\node (f) at (4,0) {$*$};
\node (ac) at (1,-.8) {$A_C$};
\node (af) at (3,.8) {$A_F$};
\node (fac) at (2,-1.6) {$*$};
\node (iaf) at (2,1.6) {$\emptyset$};
\node (acfc) at (3,-.8) {$A_{CF}$};
\node (acff) at (1,.8) {$A_{CF}$};
\draw[->] (i)--(a);
\draw[->] (a)--(f);
\draw[->] (ac)--(fac);
\draw[->] (iaf)--(af);
\draw[{Hooks[right]}->] (i)--(ac);
\draw[->>] (ac) to node[rotate=45,above=-2pt,pos=.4] {$\sim$} (a);
\draw[{Hooks[right]}->] (a) to node[rotate=45,above=-2pt,pos=.5] {$\sim$} (af);
\draw[->>] (af)--(f);
\draw[->>] (acff) to node[above=-2pt] {$\sim$} (af);
\draw[{Hooks[right]}->] (ac) to node[above=-2pt] {$\sim$} (acfc);
\draw[{Hooks[right]}->] (iaf)--(acff);
\draw[->>] (acfc)--(fac);
\end{tikzpicture}
\]
commute are called the \emph{fibrant replacement} of $A$ and the \emph{cofibrant replacement} of $A$, respectively. A \emph{bifibrant replacement} $A_{CF}$ of $A$ is the fibrant replacement of the cofibrant replacement, or, weakly equivalently, the cofibrant replacment of the fibrant replacement.
\end{defn}

The existence of the fibrant and cofibrant replacements is guaranteed by axiom 5.

\begin{prop}
In a model category $\mathcal C$:\\
\indent1.\ Cofibrations are exactly those with the LLP with respect to all acyclic fibrations.\\
\indent2.\ Acyclic cofibrations are exactly those with the LLP with respect to all fibrations.\\
\indent3.\ Fibrations are exactly those with the RLP with respect to all acyclic cofibrations.\\
\indent4.\ Acyclic fibrations are exactly those with the RLP with respect to all cofibrations.
\end{prop}

Let $A$ be an object of a model category $\mathcal C$. Since $\mathcal C$ is complete and cocomplete, we have the product $A\productop A$ and coproduct $A\amalg A$, with natural maps out of $A$ and into $A$, respectively.

\begin{defn}\label{not-smashpath}\index{cylinder object}\index{path object}\index{object!path}\index{object!cylinder}\index{left homotopic}\index{right homotopic}\index{homotopic!left}\index{homotopic!right}\index{homotopic}
A \emph{cylinder object} $A\wedge I$ and \emph{path object} $A^I$ are the objects that make the diagram
\[
\begin{tikzpicture}[scale=1.2]
\node (i) at (0,0) {$A \amalg A$};
\node (a) at (2,0) {$A$};
\node (f) at (4,0) {$A\productop A$};
\node (ac) at (1,-.8) {$A\wedge I$};
\node (af) at (3,.8) {$A^I$};
\draw[->] (i) to node[above] {$(\textup{id},\textup{id})$} (a);
\draw[->] (a) to node[below] {$(\textup{id},\textup{id})$} (f);
\draw[{Hooks[right]}->] (i)--(ac);
\draw[->>] (ac) to node[rotate=40,above=-2pt,pos=.4] {$\sim$} (a);
\draw[{Hooks[right]}->] (a) to node[rotate=40,above=-2pt,pos=.5] {$\sim$} (af);
\draw[->>] (af)--(f);
\end{tikzpicture}
\]
commute. For $f,g:A\to B$ and $h,k:B\to A$ if the cofibrant replacement of $(f,g):A\amalg A\to B$ is $A\wedge I$, then $f$ and $g$ are \emph{left homotopic}, denoted $f\stackrel\ell\sim g$. If the fibrant replacement of $(h,k):B\to A\productop A$ is $B^I$, then $h$ and $k$ are \emph{right homotopic}, denoted $g\stackrel r\sim k$. That is, $f\stackrel\ell\sim g$ and $g\stackrel r\sim k$ if the following diagram commutes.
\[
\begin{tikzpicture}[scale=1.2]
\node (i) at (0,0) {$A \amalg A$};
\node (a) at (2,0) {$B$};
\node (f) at (4,0) {$A\productop A$};
\node (ac) at (1,-.8) {$A\wedge I$};
\node (af) at (3,.8) {$B^I$};
\draw[->] (i) to node[above] {$(f,g)$} (a);
\draw[->] (a) to node[below] {$(h,k)$} (f);
\draw[{Hooks[right]}->] (i)--(ac);
\draw[->>] (ac) to node[rotate=40,above=-2pt,pos=.4] {$\sim$} (a);
\draw[{Hooks[right]}->] (a) to node[rotate=40,above=-2pt,pos=.5] {$\sim$} (af);
\draw[->>] (af)--(f);
\end{tikzpicture}
\]
If $f\stackrel\ell\sim g$ and $f\stackrel r\sim g$, then $f$ and $g$ are simply \emph{homotopic}.
\end{defn}

As before, the existence of cylinder and path objects is guaranteed by axiom 5. Also note that the smash product does not necessarily exist in a model category, we simply use the notation for intuition from topological spaces.

\begin{prop}\index{homotopy equivalence}\index{equivalence!homotopy}
Let $f,g\in \Hom_{\mathcal C}(A,B)$. \\
\indent1.\ If $A$ is cofibrant, then $f\lhom g$ implies $f\rhom g$. \\
\indent2.\ If $B$ is fibrant, then $f\rhom g$ implies $f\lhom g$. \\
\indent3.\ If $A,B$ are both fibrant, then $f$ is a homotopy equivalence iff $f$ is a weak equivalence.
\end{prop}

We use the term \emph{homotopy equivalence} to mean $f$ is homotopic to the identity map. We now revisit an earlier definition.

\begin{defn}\label{not-hocat}\index{homotopy category}\index{category!homtopy}
The \emph{homotopy category} of a model category $\mathcal C$ has the same objects as $\mathcal C$, with
\[
\Hom_{\Ho(\mathcal C)}(A,B) = \Hom_{\mathcal C}(A_{CF},B_{CF})/(\text{weak equivalences}).
\]
\end{defn}

We can equivalently mod out by homotopy equivalences. Note that we may think of the natural functor $\mathcal C\to \Ho(\mathcal C)$ as a localization inverting weak equivalences.

\subsection{Examples}

Now we consider model structures on some familiar categories.

\begin{ex}
$\Top$ is a model category:\\
\indent-\ The weak equivalences are homotopy equivalences.\\
\indent-\ The fibrations are Hurewicz fibrations (maps that have the RLP with respect to $A\hookrightarrow A\times I$).\\
\indent-\ $\Ho(\Top)$ is the homotopy category of topological spaces
\end{ex}

\begin{ex}
$\Top$ is a model category in a different way:\\
\indent-\ The weak equivalences are weak equivalences (maps that induce equivalences $\pi_i(A)\xrightarrow\cong \pi_i(B)$).\\
\indent-\ The cofibrations are retracts of attaching cells.\\
\indent-\ $\Ho(\Top)$ is the homotopy category of CW complexes.
\end{ex}

\begin{ex}
$\Ch_R$, the category of chain complexes over a positively graded ring $R$, is a model category:\\
\indent-\ The weak equivalences are quasi-isomorphisms.\\
\indent-\ The cofibrations are degree-wise monomorphisms.\\
\indent-\ $\Ho(\Ch_R)$ is the derived category of $\R$.
\end{ex}

\hrule
\section{More model categories and presentable categories}
\sectionsubtitle{\textit{Harry Smith, 2017-10-16}}

The motivation for this talk is the \emph{adjoint functor theorem}, which states that a functor $F$ of cocomplete categories is a left adjoint iff $F$ preserves colimits and satisfies the solution set condition. Having presentable categories allows us to get rid of the solution set condition.
\index{adjoint functor theorem}\index{theorem!adjoint functor}

\subsection{Vocabulary buildup}

We need to define several things before we get to presentable categories. Let $\mathcal C$ be a category.

\begin{defn}\index{filtered system}
A \emph{filtered system} is a functor $I\to \mathcal C$ from the index category such that:\\
\indent-\ every pair of objects has another object both map into, and\\
\indent-\ every pair of morphisms between the same objects has a third morphism that makes them equal.\\
That is, for every diagram without the dashed arrows, the dashed arrows exist.
\[
\begin{tikzpicture}[baseline=-.1cm]
\fill[black] (0,.5) circle (.05);
\fill[black] (0,-.5) circle (.05);
\fill[black] (1,0) circle (.05);
\draw[dashed,->,shorten >=4pt, shorten <=4pt] (0,.5)--(1,0);
\draw[dashed,->,shorten >=4pt, shorten <=4pt] (0,-.5)--(1,0);
\end{tikzpicture}
\hspace{3cm}
\begin{tikzpicture}[baseline=-.1cm]
\fill[black] (0,0) circle (.05);
\fill[black] (1,0) circle (.05);
\fill[black] (2,0) circle (.05);
\draw[dashed,->,shorten >=4pt, shorten <=4pt] (1,0)--(2,0);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0) to [out=30,in=150] (1,0);
\draw[->,shorten >=4pt, shorten <=4pt] (0,0) to [out=-30,in=-150] (1,0);
\end{tikzpicture}
\]
For $\kappa$ a cardinal and $A\subset I$ with $|A|<\kappa$ in objects and morphisms, a \emph{$\kappa$-filtered system} is one where adjoining the final object to form $A_0$ makes the diagram below commute.
\[
\begin{tikzpicture}[scale=1.3]
\node (a) at (0,1) {$A$};
\node (a0) at (0,0) {$A_0$};
\node (i) at (1,1) {$I$};
\draw[->] (a)--(i);
\draw[->] (a)--(a0);
\draw[->,dashed] (a0)--(i);
\end{tikzpicture}
\]
\end{defn}

A filtered system is a $\kappa$-filtered system for $\kappa=\aleph_0$. Also note that if $\kappa<\lambda$, then a $\lambda$-filtered system is immediately $\kappa$-filtered.

\begin{defn}\index{accessible category}\index{category!accessible}
A category $\mathcal C$ is \emph{$\kappa$-accessible} if it admits $\kappa$-filtered colimits and there is a small $\mathcal D\subset \mathcal C$ such that \\
\indent-\ every $a\in \mathcal C$ is a $\kappa$-filtered colimit of a system in $\mathcal D$, and \\
\indent-\ $\hom_{\mathcal C}(d,-)$, for $d\in \mathcal D$, preserves $\kappa$-filtetred colimits.
\end{defn}

\begin{defn}\index{presentable category}\index{locally presentable category}\index{category!presentable}
A category $\mathcal C$ is \emph{(locally) presentable} if it is $\kappa$-accessible for some $\kappa$ and cocomplete. It is \emph{$\kappa$-accessible} if $\mathcal C\to \mathcal D$, for $\mathcal D$ as above, preserves $\kappa$-filtered colimits.
\end{defn}

\begin{ex}
Here are some example of presentable categories:\\
\indent-\ $\Set$ is presentable, with $\mathcal D = \{*\}$. \\
\indent-\ $\Fun(A^{op},\Set)$ for $A$ small is presentable. Recall for $A=\Delta$ this is $\sSet$. \\
\indent-\ $\textup{Ring}(R)$, $\textup{Mod}(R)$, $\textup{Ch}(R)$ are all presentable. \\
\indent-\ quasi-coherent $\mathcal O_X$-modules over a scheme $X$ is presentable.
\end{ex}

We now return to the original motivation.

\begin{thm}{\sc[Adjoint functor theorem]} \index{adjoint functor theorem}\index{theorem!adjoint functor}\\
Let $F:\mathcal C\to \mathcal D$ be a functor of presentable 1-categories. Then: \\
\indent-\ $F$ is left adjoint iff $F$ preserves colimits, and \\
\indent-\ $F$ is right adjoint iff $F$ preserves limits and is $\kappa$-accessible for some $\kappa>\aleph_0$.
\end{thm}

\subsection{The $\infty$-Yoneda embedding}

Now let $\mathcal C,\mathcal D$ be $\infty$-categories.

\begin{defn}\index{localization}
A \emph{localization} of $\infty$-categories is a functor $F:\mathcal C\to \mathcal D$ with a fully faithful right adjoint. An \emph{accessible localization} is a localization whose right adjoint is accessible.
\end{defn}

\begin{thm}
If a 1-category $\mathcal C$ is presentable, then $\mathcal C$ is equivlent to an accessible localization of $\Fun(A^{op},\Set$, for some $A$ small.
\end{thm}

The above theorem should be taken in the context of the Yoneda embedding. That is, let the first arrow below be the cocompletion and the second arrow be the accessible localization.
\[
A\to \Fun(A^{op},\Set)\to \mathcal C.
\]
The Yoneda embedding in the $\infty$ setting uses the coherent nerve functor and the adjunction $\mathcal C[-],N_\Delta:\sSet\leftrightarrows \textup{sCat}$, for $\textup{sCat}$ the simplicially enriched category, where the hom-sets are simplicial sets. For some $K\in \sSet$, we have a map
\[
\begin{array}{r c l}
\textup{Map}:\ \mathcal C[K]^{op} \times \mathcal C[K] & \to & \textup{sCat}, \\
(X,Y) & \mapsto & \hom(\Delta\times X,Y).
\end{array}
\]
This gives maps
\[
\mathcal C[K^{op}\times K] \xrightarrow{\ \text{canonical}\ } \mathcal C[K^{op}]\times \mathcal C[K]\to \textup{sCat} \xrightarrow{\ \text{fibrant replacement}\ }\textup{Kan},
\]
where $\textup{Kan}$ is the category of Kan complexes. Adjunction given us a map $K^{op}\times K\to N_\Delta(\textup{Kan}) = \mathcal S$, the exponential adjunction. The $\infty$-Yoneda embedding then is $K\to \Fun(K^{op},\mathcal S) = P(K)$. This allows us to restate the previous theorem as follows.

\begin{thm}
An $\infty$-category $\mathcal C$ is presentable iff is it an accessible localization of $P(K)$, for some small $K$.
\end{thm}

\subsection{Relation to $\infty$-categories}

\begin{defn}\index{combinatorial model category}\index{model category!combinatorial}\index{category!combinatorial model}
A \emph{combinatorial model category} is one that \\
\indent-\ is presentable as a 1-category, and \\
\indent-\ is cofibrantly generated.
\end{defn}

A category is \emph{cofibrantly generated} if there is a set (as opposed to a class) of cofibrations generating the class of cofibrations.

\begin{ex}
The categories $\sSet$ and $\Fun(A^{op},\sSet)_{\text{proj}}$ are combinatorial model categories.
\end{ex}

\begin{thm}
An $\infty$-category $\mathcal C$ is presentable iff there exists a simplicial combinatorial model category $M$ with $\mathcal C = N_\Delta(M_{CF})$.
\end{thm}

\hrule
\section{Straightening and unstraightening}
\sectionsubtitle{\textit{Micah Darrell, 2017-10-23}}
\label{straightening}

Sources for this talk: \cite{lurie-htt}, \cite{gaitsgory-rozenblyum} Chapter I.1.4.\\

\noindent
There is an equivalence of $\infty$-categories $\Cart_{/\mathcal C}$ and $\Fun_\infty(\mathcal C^{op},\Cat_\infty)$, where $\Cart$ will be described later. The forward direction is \emph{straightening} and the backward direction is \emph{unstraightening}. The point of this is that $\Cart_{/\mathcal C}$ is a purely $(\infty,1)$-object, whereas $\Cat_\infty$ is an $(\infty,2)$-category. This identifiation will do at least two things for us:
\begin{enumerate}
\item If we want to construct a functor $F:\mathcal C^{op}\to \Cat_\infty$, we can equivalently construct a functor $p:G\to \mathcal C$ and check that $p$ is a Cartesian fibration.
\item If we have a diagram $F:\mathcal C^{op}\to \Cat_\infty$, we can ``unstraighten" this to a functor $p:G\to \mathcal C$, and then complete $\varprojlim F$ through $p$.
\end{enumerate}
\index{straightening}\index{unstraightening}

\subsection{The Grothendieck construction}

\begin{defn}\index{category over $\mathcal C$}\index{arrow}\index{cartesian arrow}
A \emph{category over $\mathcal C$} is the data of a category $F$ and a functor $p:F\to \mathcal C$. If $p:F\to \mathcal C$ is a category over $\mathcal C$, an \emph{arrow} $\varphi:d_0\to d_1$ in $F$ is called \emph{$p$-cartesian} if for any object $d'$ and any map $\lambda:d'\to d_1$ such that $F(\lambda)=g\circ f$, there exists a unique $\psi:d'\to d_0$ such that $F(\psi)=g$. That is, given the solid arrows in the diagrams below, the dotted arrow exists, keeping commutativity.
\[
\begin{tikzpicture}[yscale=1.4,xscale=1.6]
\node (a1) at (0,1) {$d'$};
\node (b1) at (0,0) {$p(d')$};
\node (c) at (1,1) {$d_0$};
\node (d) at (1,0) {$p(d_0)$};
\node (a2) at (2,1) {$d_1$};
\node (b2) at (2,0) {$p(d_1)$};
\foreach \x\y\n in {a1/b1/{}, a2/b2/{}, c/d/{}, c/a2/\varphi, b1/d/g, d/b2/f}{
  \draw[->] (\x) to node[above] {$\n$} (\y);
}
\draw[->,dashed] (a1) to node[above] {$\psi$} (c);
\draw[->] (a1)..controls +(60:.8) and +(120:.8).. node[above] {$\lambda$} (a2);
\draw[->] (b1)..controls +(-60:.8) and +(-120:.8)..(b2);
\end{tikzpicture}
\]
\end{defn}

\begin{defn}\index{fibered category}\index{category!fibered}
A \emph{fibered category} over $\mathcal C$ is a category over $\C$, $p:F\to \mathcal C$, such that for every morphism $g:U\to V$ and $|mathcal C$ and $v\in p^{-1}(V)$, there exists a Cartesian arrow $\varphi:u\to v$ such that $p(\varphi)=g$. 
\end{defn}

So in particular, there exists $u\in p^{-1}(U)$.

\subsection{The category $F(U)$}

Now we define a new category. For every object $U\in \mathcal C$, let $F(U)$ be the category whose objects are $u\in F$ such that $p(u)=U$, and morphisms are $f:u\to v$ such that $p(f)=\id_U$, so $\obj(F(U)) = p^{-1}(U) \subseteq \obj(F)$. The arrows in $F(U)$ are arrows in $F$ that $p$ sends to $\id_U$. 

\begin{rem}
Applying the axiom of choice, the functor $p:F\to \mathcal C$ is equivlent to a functor $\mathcal C^{op}\to \Cat$:
\[
\begin{array}{r c l}
\mathcal C^{op}\owns U & \mapsto & F(U)\in \Cat, \\
(g:U\to V) & \mapsto & (\text{chocie of Cartesian map in $F$}).
\end{array}
\]
Recall that a Cartesian map in $F$ is a functor $F(V)\to F(U)$. The idea is to consider the following diagram:
\[
\begin{tikzpicture}[xscale=1.5,yscale=1.3]
\node (u) at (0,0) {$U$};
\node (v) at (1,0) {$V$};
\node (fu) at (0,1) {$F(U)$};
\node (fv) at (1,1) {$F(V)$};
\node[anchor=west] at (v) {$\ \in \mathcal C$};
\foreach \x\y in {fu/u, fv/fu, fv/v}{
  \draw[->] (\x)--(\y);
}
\draw[->] (u) to node[above] {$g$} (v);
\end{tikzpicture}
\]
\end{rem}

So, $p:F\to \mathcal C$ is a Cartesian fibration if for every $(g:U\to V)\in \mathcal C$ and $v\in p^{-1}(V)$, there exists a Cartesian arrow $\varphi:u\to v$ in $F$ such that $p(\varphi)=g$. So, in particular, $u\in p^{-1}(U)$. 

\begin{ex}
The functor from spaces to the categroy of categories is a cartesian fibration:
\[
\begin{array}{r c l}
\mathcal S & \to & \Cat, \\
X & \mapsto & \Vect(X), \\
(f:X\to Y) & \mapsto & (f^*:\Vect(Y)\to \Vect(X)),
\end{array}
\]
where $f^*$ is the pullback of vector bundles.
\end{ex}

If we choose for every $f\in \mathcal C$ a Cartesian arrow in $F$, then this data specifies a functor $U\to F(U)$, with $(f:U\to V)\mapsto (F^*F(V)\to F(U))$. The punchline of this whole talk is that the $(\infty,1)$-Grothendieck construction does the same thing with Cartesian fibrations of $(\infty,1)$-categories.\\

\hrule
\section{Constructible sheaves and exit paths}
\sectionsubtitle{\textit{J\=anis Lazovskis, 2017-10-30}}

Sources for this talk: \cite{lurie-htt} Chapter 1, \cite{lurie-ha} Appendix A, \cite{groth} Section 2.

\subsection{Locally constant sheaves}

Let $X$ be a topological space and $Op(X)$ the category of open sets (and inclusions) of $X$. Recall:\\
\indent-\ $\Kan$ is the full subcategory of $\sSet$ spanned by Kan complexes (lifting property),\\
\indent-\ $\mathcal S = N(\Kan)$ is the $\infty$-category of spaces, given as the nerve.\\
The simplicial nerve $N(\mathcal C)$ of an $\infty$-category $\mathcal C$ is an $\infty$-category whenever the mapping spaces between objects of $N(\mathcal C)$ are Kan complexes.


\begin{defn}\index{presheaf}\index{sheaf}\index{stalk}
A \emph{presheaf} $\mathcal F$ on $X$ is a functor $\mathcal F:Op(X)^{op}\to \Set$. A \emph{sheaf} $\mathcal F$ on $X$ is a presheaf that satisfies the \emph{gluing condition}: given a cover $\{U_i\}_{i\in I}$ of $X$ and a collection $\{s_i\in \mathcal F(U_i)\}_{i\in I}$, 
\[
\Bigg(\mathcal F(U_i\cap U_j \hookrightarrow U_i)(s_i) = \mathcal F(U_i\cap U_j \hookrightarrow U_j)(s_j)\ \forall\ i,j\in I\Bigg)
\ \ \implies\ \ 
\Bigg(
\exists!\ s\in \mathcal F(X)\ :\ \mathcal F(U_i\hookrightarrow X)(s) = s_i\ \forall\ i\in I
\Bigg).
\]
The \emph{stalk} of $\mathcal F$ at $x\in X$ is $\mathcal F_x \colonequals \varinjlim_{x\in U} \mathcal F(U)$.
\end{defn}

This definition may be generalized to a sheaf on a category $\mathcal C$ valued in the $\infty$-category of spaces $\mathcal S$. However, this requires the deinfinition of a \emph{site} and \emph{coverage} of a category, which is not relevant here.\\

\begin{ex}
Some common examples of sheaves are:
\begin{itemize}
\item the \emph{constant sheaf} $\mathcal F(U) = A$ for some $A\in \Set$ and all $U\subseteq X$,
\item the \emph{locally constant sheaf} $\mathcal F$: every $x\in X$ has a neighborhood $U\owns x$ such that $\mathcal F|_U$ is constant.
\end{itemize}
\end{ex}

\label{not-sheaves}
Let $\Shv(X)$ be the category of sheaves on $X$, and $\Shv_{LC}(X)\subseteq \Shv(X)$ the category of locally constant sheaves on $X$ (a morphism of sheaves $\mathcal F\to \mathcal G$ is a natural transformation). 

\begin{prop}\label{lcprop}
$\Shv_{LC}(X) \cong \mathcal S_{/\Sing(X)}$, and so is an $\infty$-category. 
\end{prop}

\begin{proof} (sketch)
Remark A.1.4 of \cite{lurie-ha} gives that $\Shv(X)$ has \emph{locally constant shape} if $X$ is a paracompact topological space with the homotopy type of a CW complex. Because of this Theorem A.1.15 of \cite{lurie-ha} implies that $\Shv(X)\cong \mathcal S_{/K}$ for some fixed $K\in \mathcal S$. Since the \emph{shape} of $\Shv(X)$ can be identified with $\Sing(X)$, it follows that $K = \Sing(X)$.\\

Finally, Proposition 2.17 of \cite{groth} gives that the slice category $\mathcal S_{/\Sing(X)} \colonequals \mathcal S_{/p}$, for $p:\Delta^0\to \mathcal S$ given by $*\mapsto \Sing(X)$, is an $\infty$-category.
\end{proof}

\noindent
This proof is not very satisfying. Some new terms, for $\mathcal X$ a topos:\index{shape}\index{constant shape}\index{locally constant shape}\\
\indent-\ the \emph{shape} of $\mathcal X$ is the functor $\pi_*:\mathcal X\to \Set$ corepresented by the final object $\{*\}$ in $\mathcal X$ \\
\indent-\ $\mathcal X$ has \emph{constant shape} if $\pi_*\pi^*:\Set \to \Set$ is corepresentable \\
\indent-\ $X\in\mathcal X$ has \emph{constant shape} if $\mathcal X_{/X}$ has constant shape\\
\indent-\ $\mathcal X$ has \emph{locally constant shape} if every $X\in \mathcal X$ has constant shape\\

Now we generalize locally constant sheaves to \emph{constructible} sheaves.

\subsection{Constructible sheaves}

\begin{defn}\index{stratification}
Let $(A,\leqslant)$ be a poset. An \emph{$A$-stratification} (or simply \emph{stratification} when $A$ is clear) is a continuous map $f:X\to A$, where $A$ is given the upset topology ($x\in U$ and $x\leqslant y$ implies $y\in U$).
\end{defn}

If $X$ has a stratification, then $X$ is called a \emph{stratified space}. Given a stratification $f:X\to A$, the space $X_a = \{x\in X\ :\ f(x)=a\}$ is called a \emph{level set}. Objects $X_{>a}$, $X_{\geqslant a}$ are defined analogously.\index{level set}\index{stratified space}

\begin{ex}
Consider the following examples of $A$-stratifications.\\
\indent-\ Let $X = S^2$ and $A = \{0\leqslant 1\leqslant 2\}$. Pick a point $X_0$ and a great circle $X_0\cup X_1$ through that point.
\[
\begin{tikzpicture}
\draw (0,0) circle (1);
\draw (1,0) arc (0:-180:1 and .3);
\draw[dashed] (1,0) arc (0:180:1 and .3);
\fill[fill=black] (0,-.3) circle (.05);
\end{tikzpicture}
\]
\indent-\ Let $X$ be a simplicial complex and $A = \Z_{\geqslant 0}$. Let $X_a$ be the $a$-skeleton of $X$.
\end{ex}

\begin{defn}\label{not-constructiblesheaf}
Let $f:X\to A$ be a stratified space. An element $\mathcal F\in \Shv(X)$ is \emph{constructible} if $\mathcal F|_{X_a}\in \Shv_{LC}(X_a)$ for all $a\in A$. The category of $A$-constructible sheaves is denoted $\Shv^A(X)$.
\end{defn}

Note that $\Shv_{LC}(X)\subseteq \Shv^A(X)$ for all $A$, and $\Shv_{LC}(X) = \Shv^{\{*\}}(X)$. By Proposition A.5.9 of \cite{lurie-ha}, every element of $\Shv^A(X)$ is \emph{hypercomplete} as an object of $\Shv(X)$, which implies that $\Shv^A(X)$ is an $\infty$-category.

%\begin{proof}
%{\color{red}FINISH} use LC-sheaf inf-cat property?
%\end{proof}

\subsection{Exit path equivalence}

Now we describe $A$-constructible sheaves in a different way.

\begin{defn}\index{exit path}
An \emph{exit path} in a stratified space $f:X\to A$ is a continuous map $\gamma:[0,1]\to X$ for which there exists a pair of chains $a_1\leqslant \cdots \leqslant a_n$ in $A$ and $0=t_0\leqslant \cdots \leqslant t_n=1$ in $[0,1]$ such that $f(\gamma(t))=a_i$ whenever $t\in (t_{i-1},t_i]$.
\end{defn}

This really is a path, and so gives good intuition for what is happening. Recall that the \emph{geometric realization} of the $n$-simplex $\Delta^n$ is $|\Delta^n| = \{(t_0,\dots,t_n)\in \R^{n+1}\ :\ t_0+\cdots+t_n=1\}$. Oserving that $[0,1]\cong|\Delta^1|$, this definition may be generalized by instead considering maps from $|\Delta^n|$.

\begin{defn}
An \emph{exit path} in a stratified space $f:X\to A$ is a continuous map $\gamma:|\Delta^n|\to X$ for which there exists a chain $a_0\leqslant \cdots \leqslant a_n$ in $A$ such that $f(\gamma(t_0,\dots,t_i,0,\dots,0))=a_i$ for $t_i\neq 0$.
\end{defn}

\label{not-exitpath}
Let $\Sing^A(X)\subseteq \Sing(X)$ be the category of exit paths on $f:X\to A$. If $f:X\to A$ is \emph{conically stratified}, then $\Sing^A(X)$ is an $\infty$-category (Theorem A.6.5 in \cite{lurie-ha}). This means that every $x\in X$ has a neighborhood that looks like $Z\times C(Y)$, for appropriate topological spaces $Z,Y$ (and $C(Y)$ the cone of $Y$).

\begin{lem}
If $X$ has locally singular shape, then $\Shv_{LC}(X)\cong \Fun(\Sing(X),\mathcal S)$.
\end{lem}

It is sufficient for $X$ to be locally contractible for $X$ to have locally singular shape. This may be viewed as straightening and unstraightening (see Section \ref{straightening}).\\

This lemma combines Proposition \ref{lcprop} above and Theorem A.1.15 in \cite{lurie-ha}, as $\Sing(X)$ is the shape of $\Shv(X)$. Citing directly from Section A.6 of \cite{lurie-ha}, the idea now is that a sheaf $\mathcal F\in \Shv_{LC}(X)$ can be interpreted as a functor which assigns to each $x\in X$ the stalk $\mathcal F_x\in \mathcal S$ and to each path joining $x$ to $y$ the homotopy equivalence $\mathcal F_x\cong \mathcal F_y$. This generalizes to:

\begin{thm}
If $X$ has locally singular shape and $f:X\to A$ is conically stratified, then $\Shv^A(X) \cong \Fun(\Sing^A(X),\mathcal S)$.
\end{thm} 

We now follow Construction A.4.17 and its variants in Section A.9 of \cite{lurie-ha}. It should be noted the first source uses $\mathcal P(\mathcal U(X))$ to mean presheaves on open set of $X$ and the second uses $\mathcal P(\mathcal B(X)$ to mean presheaves on open $F_\sigma$ subsets of $X$, but we conflate the notation here.
\[
\begin{array}{r c l l}
(U,Y) & \mapsto & \Fun_{\Sing(X)}(\Sing(U),Y)\\
\theta\ :\ Op(X)^{op} \times \sSet_{/\Sing(X)} & \to & \sSet \\[10pt]
\theta|_{bifib}\ :\ Op(X)^{op}\times \mathbf A_X^o & \to & \Kan & \text{take bifibrant objects}\\[10pt]
N(\theta|_{bifib})\ :\ N(Op(X)^{op})\times N(\mathbf A_X^o) & \to & \mathcal S & \text{take nerve} \\[10pt]
\psi \ :\ N(\mathbf A_X^o) & \to & p\Shv(X) & \text{equivalently}\\[10pt]
\psi \ :\ \Fun(\Sing^A(X), \mathcal S) & \to & p\Shv(X) & \text{Notation A.9.1}
\end{array}
\]

Above, we let $\mathbf A_X$ denote $\sSet_{/\Sing(X)}$ and $\mathbf A_X^o$ the bifibrant objects of $\mathbf A_X^o$. The bifibrant objects of $\sSet$ (with the usual model category structure) are Kan complexes. The map $\psi$ now factors in the following way:
\[
\begin{tikzpicture}[yscale=1.8,xscale=3.2]
\node (cs) at (0,0) {$\Shv^A(X)$};
\node (s) at (1,0) {$\Shv(X)$};
\node (n) at (0,1) {$\Fun(\Sing^A(X),\mathcal S)$};
\node (ps) at (1,1) {$p\Shv(X)$};
\draw[->] (n) to node[above] {$\psi$} (ps);
\draw[->,dashed] (s)--(ps);
\draw[->] (n) to node[auto] {if c.} (s);
\draw[->,dashed] (cs)--(s);
\draw[->] (n) to node[left] {if l.s.s.} (cs);
\end{tikzpicture}
\]
If the $A$-stratification of $X$ is conical, then the map $\psi$ factors through $\Shv(X)$. If further $X$ has locally singular shape, then this new map factors through $\Shv^A(X)$. Theorem A.9.3 of \cite{lurie-ha} describes why this last map is an equivalence.\\

\hrule
\section{Symmetric monoidal infinity categories}
\sectionsubtitle{\textit{Maximilien P\'eroux, 2017-11-06}}

Sources for this talk: \cite{groth}, \cite{lurie-htt}.

\subsection{Ordinary categories}

The following is the main example to keep in mind.

\begin{ex}
Let $(M,\otimes,\mathbf 1)$ be a monoidal category. Define a new category $M^\otimes$ with:
\begin{center}
\begin{tabular}{r l}
objects: & $(M_1,\dots,M_n)$ for $n\geq 0$, $M_i\in M$, \\
morphisms: & $(M_1,\dot,M_n) \xrightarrow{\ \alpha,\{f_i\}\ } (L_1,\dots,L_k)$, where $\alpha:[k]\to [n]$ in $\Delta$ and $f_i: M_{\alpha(i-1)+1}\otimes \cdots \otimes M_{\alpha(i)} \to L_i$.
\end{tabular}
\end{center}
This gives a functor which completely describes the monoidal structure of $M$, called the \emph{Grothendieck opfibration}:
\[
\begin{array}{r c l}
p\ :\ M^\otimes & \to & \Delta^{op}, \\
(M_1,\dots,M_n) & \mapsto & [n], \\
(\alpha,\{f_i\}) & \mapsto & \alpha.
\end{array}
\]
\end{ex}
\index{Grothendieck opfibration}\index{opfibration}

\begin{defn}\label{catfiber}\index{fiber}
Let $p:\mathcal C\to \mathcal D$ be a functor. The object $\mathcal C_d$ is the fiber $d\in \mathcal D$ over $p$, defined as the pullback
\[
\begin{tikzpicture}
\node (s) at (0,0) {$*$};
\node (cd) at (0,1) {$\mathcal C_d$};
\node (c) at (1,1) {$\mathcal C$};
\node (d) at (1,0) {$\mathcal D$};
\foreach \x\y in {cd/s, cd/c, c/d, s/d}{
  \draw[->] (\x)--(\y);
}
\draw (.7,.3)--(.5,.3) (.7,.3)--(.7,.5);
\end{tikzpicture}
\]
\end{defn}

We would like the asignment $d\mapsto \mathcal C_d$ to be functorial, but it is not yet functorial. 

\begin{defn}\index{coCartesian lift}\index{lift}
Given a map $c_1\xrightarrow{\ f\ } c_2$ in $\mathcal C$, it is a \emph{$p$-coCartesian lift} of $\alpha:d_1\to d_2$ in $\mathcal D$ if $p(f) = \alpha$ and 
\[
\begin{tikzpicture}[yscale=1.4,xscale=1.6,baseline=.6cm]
\node (a1) at (0,1) {$c_1$};
\node (b1) at (0,0) {$d_1$};
\node (c) at (1,1) {$c_2$};
\node (d) at (1,0) {$d_2$};
\node (a2) at (2,1) {$c_3$};
\node (b2) at (2,0) {$d_3$};
\foreach \x\y\n in {a1/c/f, b1/d/\alpha, d/b2/{\forall \beta}}{
  \draw[->] (\x) to node[above] {$\n$} (\y);
}
\draw[->,dashed] (c) to node[above] {$\exists!g$} (a2);
\foreach \x\y\n in {a1/b1/{p}, a2/b2/{}, c/d/{p} }{
  \draw (\x) to node[left] {$\n$} (\y);
}
\draw[->] (a1)..controls +(60:.8) and +(120:.8).. node[above] {$\forall\ h$} (a2);
\draw[->] (b1)..controls +(-60:.8) and +(-120:.8).. node[below] {$\forall\ \gamma$} (b2);
\end{tikzpicture}
\hspace{1cm}
\iff
\hspace{1cm}
\begin{tikzpicture}[xscale=1.5,yscale=1.3,baseline=.6cm]
\node (s) at (0,0) {$\mathcal C_{c_1/}$};
\node (cd) at (0,1) {$\mathcal C_{f/}$};
\node (c) at (1,1) {$\mathcal D_{p(c_1)/}$};
\node (d) at (1,0) {$\mathcal C_{p(c_1)/}$};
\foreach \x\y in {cd/s, cd/c, c/d, s/d}{
  \draw[->] (\x)--(\y);
}
\draw (.7,.3)--(.5,.3) (.7,.3)--(.7,.53);
\end{tikzpicture}
\]
commute.
\end{defn}

\begin{lem}
Given $c\xrightarrow{\ f'\ }c'$ and $c\xrightarrow{\ f''\ }c''$ $p$-coCartesian lifts of $\alpha$, then $c'\xrightarrow{\ \cong\ }c''$. 
\end{lem}

\begin{defn}\label{abang}
A functor $p:\mathcal C\to \mathcal D$ is a Grothendieck opfibration if for all $c_1\in \mathcal C$ and any morphism $\alpha\in \mathcal D$ with domain $p(c_1)$ there is a $p$-coCartesian lift $c_1\to c_2$ of $\alpha$. Given such $\alpha:p(c_1)=d_1\to d_2$ in $\mathcal D$, we get a new functor
\[
\begin{array}{r c l}
\alpha!\ :\ \mathcal C_{d_1} & \to &\mathcal C_{d_2}, \\
c_1 & \mapsto & c_2,
\end{array}
\]
where $c_2$ is the codomain of a lift of $\alpha$.
\end{defn}

Note that $\mathcal D\to \Cat$, given by $d\mapsto \mathcal C_d$ is still not functorial. This can be seen by considering $d_1\tov \alpha d_2 \tov \beta d_3$. Going back to the main example $M^\otimes$, we have that the fiber $M^\otimes_{[n]}$ is canonically equivalent to $(M_{[1]}^\otimes)^{\times n}$. 

\begin{defn}\index{Segal maps}\index{Segal conditions}
Consider the assignment
\[
\iota_i = \left(
\begin{array}{r c l}
[1] & \to & [n], \\
0 & \mapsto & i-1, \\
1 & \mapsto & i,
\end{array}
\right)^{op}
\ \ \ 
\begin{array}{c}
\phantom a \\ \rightsquigarrow \\ \phantom a
\end{array}
\ \ \ 
\begin{array}{r c l}
(\iota_i)!\ :\ M_{[n]}^\otimes & \to & M_{[1]}^\otimes = M, \\
(M_1,\dots,M_n) & \mapsto & M_i.
\end{array}
\]
These induce new maps $\sigma = ((\iota_1)!,\dots,(\iota_n)!): M_{[n]}^\otimes \times \cdots \times M_{[1]}^\otimes$, called \emph{Segal maps}.
\end{defn}

The \emph{Segal conditions} are the conditions which, when satisfied, maked $\sigma$ an equivalence. This approach can be generalized: given a Gorthendieck opfibration $\mathcal C\to \Delta^{op}$ respecting the Segal conditions, we get a monoidal category $M = \mathcal C_{[1]}$. Define $\otimes:M\times M\to M$ by letting $d_1:[2]\to[1]$ be the face map, so then
\[
\begin{tikzpicture}
\node (a) at (0,0) {$\otimes\ :\ M\times M$};
\node[anchor=west] (b) at ($(a.east)+(0:1)$) {$\mathcal C_{[2]}$};
\node[anchor=west] (c) at ($(b.east)+(0:1)$) {$\mathcal C_{[1]} = M$.};
\draw[->] (b) to node[above] {$\cong$} node[below] {$\sigma$} (a);
\draw[->] (b) to node[above] {$(d_1)!$} (c);
\end{tikzpicture}
\]
The unit is defined by $[0]\to [1] \rightsquigarrow *\to \mathcal C_{[1]} = M$. To get a \emph{symmetric} monoidal category, change the category $\Delta^{op}$ to $\Fin_*$, the category of finite pointed sets.

\subsection{The $\infty$-setup}

\begin{defn}\index{coCartesian lift}\index{coCartesian fibration}
Let $p:\mathcal C\to \mathcal D$ be a functor of $\infty$-categories. A map $c_1\tov f c_2$ is a \emph{coCartesian lift} of $\alpha = p(f)$ in $\mathcal D$ if $\mathcal C_{f/}\to \mathcal C_{c_1/} \times_{\mathcal D_{p(c_1)/}} \mathcal D_{p(f)/}$ is a trivial Kan fibration. A \emph{coCartesian fibration} $p:\mathcal C\to \mathcal D$ is:
\begin{enumerate}
\item an inner fibration, and 
\item for all $c_1\in \mathcal C$, for all $\alpha = p(c_1):d_1\to d_2$ in $\mathcal D$, there is a $p$-coCartesian lift $c_1\tov fc_2$.
\end{enumerate}
\end{defn}

\begin{defn}\index{monoidal $\infty$-category}\index{category!monoidal $\infty$-}
A \emph{monoidal $\infty$-category} is a coCartesian functor $p:M^\otimes \to N(\Delta^{op})$ that respects the Segal conditions.
\end{defn}

This definition implies that $\Ho(M^\otimes)$ inherits a monoidal structure. 

\begin{ex}
Here are two examples of monoidal $\infty$-categories.
\begin{enumerate}
\item Given $M$ an ordinary monoidal category, $M^\otimes \tov p \Delta^{op}$, take the nerve $N(p):N(M^\otimes) \tov{N(p)} N(\Delta^{op})$.
\item A nice model category gives an $\infty$-category via the coherent nerve $N_\Delta$. That is, $N_\Delta(M_{CF}) \to N_\Delta(\Delta^{op}) = N(\Delta^{op})$.
\end{enumerate}
\end{ex}

\subsection{Constructing an algebra}

Given $M^\otimes \to \Delta^{op}$, a section $A:\Delta^{op} \to M^\otimes$ would maybe define an algebra. Consider the Segal map and face map:
\[
\begin{array}{r c l}
M_{[n]}^\otimes & \to & M\times \cdots \times M, \\
A_{[n]} & \to & (A_{[n]}^1,\dots,A_{[n]}^n),
\end{array}
\hspace{2cm}
\begin{array}{r c l}
d_1\ :\ [2] & \to & [1], \\
(A_{[2]}^1, A_{[2]}^2) & \to & A_{[1]}^1, \\
(A_{[2]}^1, A_{[2]}^2) & \to & A_{[2]}^1 \otimes A_{[2]}^1.
\end{array}
\]
We would like $A_{[2]}^1 \cong A_{[2]}^2 \cong A_{[2]}^1$. The $p$-coCartesian lift would be $A_{[2]}^1\otimes A_{[2]}^1 \to A_{[2]}^1$. 

\begin{defn}\index{convex morphism}
A morhpism $\alpha:[n]\to [k]$ in $\Delta$ is \emph{convex} if it is a monomorphism and $\im(\alpha) = [\alpha(0),\alpha(n)]$ is an interval.
\end{defn}

In particular, the $\iota_i$ are convex. \\ 

\hrule
\section{Higher category theory in algebraic geometry}
\sectionsubtitle{\textit{Micah Darrell, 2017-11-13}}

Sources for this talk: \cite{bzfn}.\\

In \cite{bzfn}, it is shown that for $X,Y,Z$ \emph{perfect stacks}, $QC(X\times_Z Y) \cong QC(X)\otimes_{QC(Z)} QC(Y)$. Then \emph{Hochshild homology} may be restated as
\[
HH(X) \colonequals QC(X) \otimes_{QC(X)\otimes QC(X)} QC(X) \cong QC(X\times_{X\times X} X) \cong QC(\Map(S^1,X)) = QC(\mathcal L X),
\]
where $\mathcal LX$ is the \emph{loop space} on $X$. Recall that we also have $\pi QC(X) = \mathcal D(X)$, the \emph{derived category} of $X$.

\subsection{Tensor of presentable $\infty$-categories}

\begin{defn}\label{prl}
$\Pr^L$ is the sub-$(\infty,1)$-category of presentable $\infty$-categories whose morphisms are colimit preserving functors.
\end{defn}

The ``L" is for ``left-adjoint." If $\mathcal C,\mathcal D\in \Pr^L$, then there is an $\infty$-category $\mathcal C\otimes \mathcal D\in \Pr^L$ with thte property that if $E\in \Pr^L$ and $F:\mathcal C\times \mathcal D\to E$ is a functor that preserves colimits separately in each variable, then $F$ factors through $\mathcal C\times \mathcal D\to \mathcal C\otimes \mathcal D$.\\

For \emph{affine derived sheaves} $A$, we set $QC(A) = \Mod_A$. Then for a \emph{derived stack} $X = \colim_{U\in \Aff_{/X}} U$, set $QC(X) = \lim_{U\in \Aff_{/X}} QC(U)$. Also note that $\pi:\infty cat\to CAT$ is a right adjoint to the nerve functor $N:CAT\to \infty cat$.

\begin{defn}\index{perfect module}\label{perfectmodule}
Let $Z$ be a derived ring. An $A$-module $M$ is called \emph{perfect} if it is the smallest subcategory of $\Mod_A$ which contains $A$ and is closed under finite colimits and retracts. The category of such modules is denoted $\Perf(A)$.
\end{defn}

Then for a general \emph{derived stack} $X$, an element $M\in QC(X)$ is called \emph{perfect} is for any $F:U\to X$, for $U$ affine, we have $f^*M\in \Perf(U)$. We call a derived stack a \emph{perfect stack} if $QC(X) = \Ind\Perf(X)$, where $\Ind$ means take the inductive limits. Note that:
\begin{itemize}
\item $QC(X)$ has finite colimits, so if $X$ is perfect, then $QC(X)$ is presentable.
\item If $X$ is a perfect stack, then $M\in \Perf(X) \iff (\Hom_{QC(X)}(M,-)$ commutes with colimits).
\item Quasi-compact schemes with affine diagonal are perfect stacks.
\end{itemize}

\subsection{BZFN proof}

To prove the given statement atthe beginning, the authors first show that it holds for compact objects (equivalently, perfect objects). They use the exterior product
\index{adjoint functor theorem}\index{theorem!adjoint functor}
\[
\begin{array}{r c l}
\boxtimes\ :\ QC(X)^c \times QC(X)^c & \tov \sim & QC(X\times Y)^c, \\
(M,N) & \mapsto & P_1^*M\otimes P_2^*N.
\end{array}
\]
To prove the general case, we use a corollary of the \emph{adjoint functor theorem}, which says $(\Pr^L)^{op} \cong \Pr^R$. We know that $QC(X\times Y) \cong QC(X)\otimes QC(Y)$ in the regular category setting. Begin with
\[
\begin{tikzpicture}
\node (0) at (0,0) {$X\times_Z Y$};
\node[anchor=west] (1) at ($(0.east)+(0:1)$) {$X\times Y$};
\node[anchor=west] (2) at ($(1.east)+(0:1)$) {$X\times Z\times Y$};
\node[anchor=west] (3) at ($(2.east)+(0:1)$) {$X\times Z\times Z\times Y$};
\node[anchor=west] (4) at ($(3.east)+(0:1)$) {$\cdots$,};
\draw[->] (0) to node[above] {$\pi$} (1);
\draw[->,transform canvas={yshift=2pt}] (1)--(2);
\draw[->,transform canvas={yshift=-2pt}] (1)--(2);
\draw[->] (2)--(3);
\draw[->,transform canvas={yshift=4pt}] (2)--(3);
\draw[->,transform canvas={yshift=-4pt}] (2)--(3);
\draw[->,transform canvas={yshift=6pt}] (3)--(4);
\draw[->,transform canvas={yshift=2pt}] (3)--(4);
\draw[->,transform canvas={yshift=-2pt}] (3)--(4);
\draw[->,transform canvas={yshift=-6pt}] (3)--(4);
\end{tikzpicture}
\]
then apply $QC$, which is a contravariant functor, to get
\[
\begin{tikzpicture}
\node (0) at (0,0) {$QC(X\times_Z Y)$};
\node[anchor=west] (1) at ($(0.east)+(0:1)$) {$QC(X\times Y)$};
\node[anchor=west] (2) at ($(1.east)+(0:1)$) {$QC(X\times Z\times Y)$};
\node[anchor=west] (3) at ($(2.east)+(0:1)$) {$QC(X\times Z\times Z\times Y)$};
\node[anchor=west] (4) at ($(3.east)+(0:1)$) {$\cdots$.};
\draw[<-] (0) to node[above] {$\pi^*$} (1);
\draw[<-,transform canvas={yshift=2pt}] (1)--(2);
\draw[<-,transform canvas={yshift=-2pt}] (1)--(2);
\draw[<-] (2)--(3);
\draw[<-,transform canvas={yshift=4pt}] (2)--(3);
\draw[<-,transform canvas={yshift=-4pt}] (2)--(3);
\draw[<-,transform canvas={yshift=6pt}] (3)--(4);
\draw[<-,transform canvas={yshift=2pt}] (3)--(4);
\draw[<-,transform canvas={yshift=-2pt}] (3)--(4);
\draw[<-,transform canvas={yshift=-6pt}] (3)--(4);
\end{tikzpicture}
\]
The general term is $QC(X)\otimes QC(Z)\otimes \cdots \otimes QC(Z) \otimes QC(Y)$. Note that this is just the 2-sided \emph{bar construction}.\\

\hrule
\section{Spectra}
\sectionsubtitle{\textit{Haldun \"Ozg\"ur Bay\i nd\i r, 2017-11-20}}

Sources for this talk: \cite{groth} Section 5.2\\

\subsection{Reviewing spectra}\index{prespecturm}\index{smash product}\label{spectrum}

Let $E$ be a \emph{prespectrum}, with $E_n\in \sSet_*$ and maps $\sigma_n:S^1\wedge E_n \to E_{n+1}$. Recall the \emph{smash product} is given by $S^1\wedge E_n = S^1\times E_n / S^1\vee E_n$. There suspension and loop space are adjoints
\[
(\Sigma,\Omega)\ :\ \sSet_* \rightleftarrows \sSet_*.
\]

\begin{defn}
A prespectrum $E$ is a \emph{spectrum} if the adjoint maps $E_n\to \Omega E_{n+1}$ are homeomorphisms.
\end{defn}

We now have a functor
\[
\begin{tikzpicture}
\node (ss) at (0,0) {$\sSet_*$};
\node (sp) at (3,0) {$\Sp$,};
\node (psp) at (1.5,-1) {$\preSp$};
\foreach \x\y in {ss/sp, ss/psp, psp/sp}{
  \draw[->] (\x)--(\y);
}
\node (ss1) at ($(ss)+(225:.8)$) {$X$};
\node (psp1) at ($(psp)+(270:1)$) {$E_n'$};
\node (sp1) at ($(sp)+(315:.8)$) {$E_n$};
\node[anchor=north] (psp2) at ($(psp1.south)+(270:.5)$) {$\Sigma^n X$};
\node[anchor=west] (sp2) at ($(sp1.east)+(0:.5)$) {$\colim_{\sigma_i}\Omega^i\Sigma^i E_n'$};
\node[rotate=90] at ($(psp1.south)!.5!(psp2.north)$) {$=$};
\node at ($(sp1.east)!.5!(sp2.west)$) {$=$};
\foreach \x\y in {ss1/psp1, psp1/sp1}{
  \draw[|->] (\x)--(\y);
}
\node[anchor=east] at (ss.west) {$\vphantom{a_*}\Sigma^\infty\ :$};
\end{tikzpicture}
\]
where $\Sp$ is the category of spectra. This induces a smash product on the homotopy category $\Ho(\Sp)$, as well as on $\Sp$.

\begin{rem}
There are new models of spectra, which are all equivalent: LMS (Lewis--May--Steinberger), EKMM (Elmendorf--Kriz--Mandell--May), symmetric spectra (Hovey--Shipley--Smith), MMSS (Mandell--May--Schwede--Shipley).
\end{rem}

To get a (commutative) ring spectrum, we would like the map $E\wedge E\to E$ to be unital and associative (and commutative).

\begin{rem}
Spectra are very useful. For $E,F$ (commutative) ring spectra, we have
\begin{itemize}
\item $E_\bullet(-)$ is a homology theory,
\item $E_\bullet F = \pi_\bullet E\wedge F$ is a (commutative) ring with power operations, 
\item $E^\bullet(-)$ is a cohomology theory,
\item $E^n F = \pi_{-n}\Map_{\Sp}(F,E)$ incorprates the mapping space, and
\item $E^n X = \pi_{-n}\Map_{\Sp}(\Sigma^\infty X_*,E)$ are (commutative) rings, for $X\in \sSet$.
\end{itemize}
We also get algebra structures and power operations on spectral sequences that allow us to calculate (for example) THH and $E_\bullet F$. 
\end{rem}

\subsection{Spectra in $\infty$-categories}

All $\infty$-categories will be presentable.

\begin{defn}\index{triangle}\index{exact triangle}\index{coexact triangle}\label{exacttriangles}
Let $X,Y,Z$ be objects in a category $\mathcal C$. A \emph{triangle} in $\mathcal C$ is a commutative square 
\[
\begin{tikzpicture}[scale=1.7]
\node (0) at (0,0) {0};
\node (x) at (0,1) {$X$};
\node (z) at (1,0) {$Z$,};
\node (y) at (1,1) {$Y$};
\draw[->] (x)--(0);
\draw[->] (0)--(z);
\draw[->] (x) to node[above] {$f$} (y);
\draw[->] (y) to node[right] {$g$} (z);
\draw[->] (x) to node[fill=white] (h) {$h$} (z);
\draw[-Implies,double distance=2pt,shorten >=2pt] (h)--(0);
\draw[-Implies,double distance=2pt,shorten >=2pt] (h)--(y);
\end{tikzpicture}
\]
with $g\circ f \simeq 0$ and $h\simeq 0$. If the square is a pullback, then the triangle is \emph{exact} (forming the category $\mathcal C^\Omega$), and if it is a pushout, then the triangle is \emph{coexact} (forming the category $\mathcal C^\Sigma$).
\end{defn}

\begin{ex}
For example, let $X = S^1$ the circle and $Y=D^2$ the disk. Then we have $Z = S^2$, giving the coexact triangle 
\[
\begin{tikzpicture}[scale=1.7]
\node (0) at (0,0) {$*$};
\node (x) at (0,1) {$S^1$};
\node (z) at (1,0) {$S^2$.};
\node (y) at (1,1) {$D^2$};
\draw[->] (x)--(0);
\draw[->] (0)--(z);
\draw[->] (x)--(y);
\draw[->] (y)--(z);
\end{tikzpicture}
\]
\end{ex}

Let $\mathcal C$ be a finitely (co)complete pointed $\infty$-category. Then we hve evaluation maps
\[
\begin{array}{r c l}
\ev_{(0,0)}\ :\ \mathcal C^\Sigma & \to & \mathcal C, \\
\ev_{(1,1)}\ :\ \mathcal C^\Omega & \to & \mathcal C,
\end{array}
\]
both of which are acyclic Kan fibrations. Using these we can try to define suspensions with the diagram
\[
\begin{tikzpicture}[scale=1.7]
\node (0) at ($(x)+(270:1)$) {$\Delta^0$};
\node (x) at (0,1) {$\left(\begin{array}{c}\text{contractible}\\\text{object}\end{array}\right)$};
\node[anchor=west] (y) at ($(x.east)+(0:.7)$) {$\Map (\mathcal C,\mathcal C^\Sigma)$};
\node (z) at ($(y)+(270:1)$) {$\Map(\mathcal C,\mathcal C)$.};
\draw[->] (x)--(0);
\draw[->] (0) to node[above] {$\id_{\mathcal C}$} (z);
\draw[->] (x)--(y);
\draw[->] (y) to node[right] {$\ev_{(0,0)}$} (z);
\end{tikzpicture}
\]
We then get maps $\Sigma:\mathcal C\to \mathcal C^2\to \mathcal C$ and $\Omega:\mathcal C\to \mathcal C^\Omega\to \mathcal C$, giving an adjunction $(\Sigma,\Omega):\mathcal C \leftrightarrows \mathcal C$. 

\begin{defn}
A pointed, (co)complete category $\mathcal C$ is \emph{stable} if every triangle is exact iff it is coexact.
\end{defn}

\begin{thm}
The following are equivlent:
\begin{enumerate}
\item $(\Sigma,\Omega)$ is an equivalence of categories.
\item $\mathcal C$ is stable. 
\item Every commutative squares is a pullback iff it is a pushout.
\end{enumerate}
\end{thm}

\index{spectrum}
To define a \emph{spectrum}, we begin with a functor $X:N(\Z\times \Z) \to \mathcal C$ such that $X(i,j) = 0$ for all $i\neq j$. This gives a commutative diagram.
\[
\begin{tikzpicture}[scale=1.4]
\node (xn-1) at (0,0) {$X_{n-1}$};
\node (xn) at (1,1) {$X_n$};
\node (xn+1) at (2,2) {$X_{n+1}$};
\node (01) at (0,1) {0};
\node (02) at (1,0) {0};
\node (03) at (1,2) {0};
\node (04) at (2,1) {0};
\node[rotate=90] at ($(xn-1)+(225:.7)$) {$\ddots$};
\node[rotate=90] at ($(xn+1)+(45:.7)$) {$\ddots$};
\foreach \x\y in {xn-1/01, xn-1/02, 01/xn, 02/xn, xn/03, xn/04, 03/xn+1, 04/xn+1}{
  \draw[->] (\x)--(\y);
}
\end{tikzpicture}
\]
where $X_n = X(n,n)$. We also have maps $\alpha_{n-1}:\Sigma X_{n-1}\to X_n$ and $\beta_n:X_n\to \Omega X_{n+1}$. Then we say that $X$ is a \emph{spectrum below $n$} if $\beta_m$ is a weak equivalence for all $m<n$, and simply a \emph{spectrum} if $\beta_n$ is a weak equivalence for all $n$. \\

The next step is to define a functor $\preSp(\mathcal C)\to \Sp(\mathcal C)$, which can be seen in Proposition 5.23 of \cite{groth}.\\

\hrule
\section{Topoi and $\infty$-topoi}
\sectionsubtitle{\textit{Adam Pratt, 2017-11-27}}

Sources for this talk: \cite{joyal-tierney}, \cite{maclane-moerdijk}, \cite{baez-shulman}, \cite{cisinski}, \cite{lurie-htt}.

\subsection{Elementary topoi}

The view to keep is that elementary topoi generalize sets and $\infty$-topoi generalize spaces. In topological spaces, one can think in terms of \emph{locales} and \emph{frames}. Every topological space has a frame of open sets $\Op(X)$. 

\begin{defn}\index{frame}\index{meet}\index{join}\label{meetjoin}
A \emph{frame} is a poset that has all small \emph{joins} ($\vee$) and finite \emph{meets} ($\wedge$) such that $X\wedge (\bigvee_i Y_i) = \bigvee_i(X\wedge Y_i)$.
\end{defn}

We can think of $\vee$ as $\cup$ and $\wedge$ as $\cap$, though they are not exactly the same. Note that the category of locales is the opposite of the category of frames. 

\begin{rem}\index{subobject classifier}
The category $\Set$ of sets has the following properties:
\begin{enumerate}
\item Has finite limits and all colimits
\item Is cartesian closed (has product, exponential object $X^Y$, and terminal object $\emptyset$)
\item Has a subobject classifier $\subseteq$
\end{enumerate}
\end{rem}

\begin{defn}\index{topos!elementary}\index{elementary topos}
An \emph{elementary topos} is a category which has the following properties:
\begin{enumerate}
\item Has finite limits
\item Is cartesian closed
\item Has a subobject classifier
\end{enumerate}
\end{defn}

\begin{ex}\index{Heyting algebra}
$\Set$ is an elementary topos. A \emph{Heyting algebra} is also an elementary topos - this is a poset that has the following properties:
\begin{enumerate}
\item Products are meets
\item Is cartesian closed
\item Subobject classifier is $\implies$, with $((c\wedge a)\subseteq b) \iff ((c\subseteq a) \implies b)$
\end{enumerate}
A Heyting algebra is the weakest setting to do logic (that is, has modus ponens, or $((P\implies Q)\wedge P) \implies Q$). Infact, a frame is equivalent to a complete Heyting algebra, as it has all small joins.
\end{ex}

\subsection{Sheaves}

Recall that a \emph{presheaf} of sets on a category $\mathcal C$ is a functor $F:\mathcal C^{op}\to \Set$, and the category of presheaves is $\preShv(\mathcal C) = [\mathcal C^{op},\Set]$.

\begin{ex}\index{presheaf}
$\preShv(\mathcal C)$ is an elementary topos.
\end{ex}

\begin{defn}\index{site}\index{Grothendieck topology}\index{Grothendieck topos}
A \emph{site} $(\mathcal C,J)$ is a small category $\mathcal C$ equipped with a \emph{Grothendieck topology} $J$. A \emph{Grothendieck topos} is a category equivalent to the ctegory of sheaves on some site.
\end{defn}

\begin{ex}
$\Set$ is a Grothendieck topos. It is a category of sheaves on a point.
\end{ex}

Note that $\Set$ is terminal in $\ShTopos$, the 2-category of topoi. The unique morphism is the global sections functor.

\begin{thm}
A Grothendieck topos is an elementary topos that is comploete (has all small colimits) and has a small generating set.
\end{thm}

\begin{rem}
Note that:
\begin{itemize}
\item $\Set$ is cocomplete and any nonempty set generates $\Set$.
\item Any scheme can be seen as a (locally ringed) Grothendieck topos.
\item The locale axioms are the Grothendieck topos in a 0-category sense. Moreover, any locale is a $(0,1)$-Grothendieck topos and a Heyting algebra is a $(0,1)$-elementary topos.
\end{itemize}
\end{rem}

\subsection{The $\infty$ setting}

Now we move from sets to spaces. Recall that $\infGrpd$, the category of $\infty$-groupoids, is presentable, colimits are universal, and has small ``self-reflection" (see \cite{lurie-htt} for more on this).

\begin{defn}{\sc[Giraud--Rezk--Lurie]}\index{topos!$(\infty,1)$-}\\
An \emph{$(\infty,1)$-topos} is a presentable $(\infty,1)$-category with universal colimits and an \emph{object classifier}, which is an object $c$ such that $c\mapsto\textup{Core}(\mathcal C_{/c})$ is a representable $\infty$-functor $\mathcal C^{op}\to \infGrpd$.
\end{defn}

\begin{ex}
$\infGrpd$ is terminal, with the unique morphism being the global sections functor.
\end{ex}

These objects also appear in \emph{spectral algebraic geometry}, which studies spectral schemes and schemes built from $E_\infty$ rings. Note that structure sheaves on $E_\infty$ give us $\infty$-topoi. Moreover, elementary $\infty$-topoi give categorical semantics for $\infty$-topoi.\\



\hrule
\section{$K$-theory}
\sectionsubtitle{\textit{Jack Hafer, 2017-12-04}}

Sources for this talk: \cite{weibel} Sections II.9 and IV.8, \cite{bgt}.\\

The goal of this talk will be to understand the statement ``$K:\Cat_\infty^{perf} \to S_\infty$ is additive invariant," where the superscript $perf$ means idempotent-complete categories. The statement may also be viewed as saying that $K$ inverts Morita eqivalences, preserves filtered colimits and satisfies Waldhausen additivity. A more ambitious goal would be to show that this is a universal such functor.

\subsection{Waldhausen categories}

\begin{defn}\label{arrows}\index{category with cofibrations}\index{category!with cofibrations}
A \emph{category with cofibrations} (cofibration indicated by $\rightarrowtail$) is a category satisfying:
\begin{enumerate}
\item every isomorphism is a cofibration,
\item there exists a 0-object with $0\rightarrowtail A$, and
\item if $A \cofib A$ and $A to C$, then $B\cup_A C$ exists and $C\cofib B\cup_A C$.
\end{enumerate}
\end{defn}

\noindent
Here $B\cup_A C$ indicates the pushout of $B$ and $C$ with respect to $A$. In a category with cofibrations we then have that $B\sqcup C = B\cup_0 C$ and $B/A = B\cup_A 0$. Reall a \emph{cofiber sequence} is a sequence $A\cofib B\twoheadrightarrow C$. \index{pushout}\index{cofiber sequence}

\begin{defn}\index{category with weak equivalences}\index{category!with weak equivalences}
A \emph{category with weak equivalences} (weak equivalence indicated by $\stackrel\sim\longrightarrow$) is a category satisfying:
\begin{enumerate}
\item every isomorphism is a cofibration,
\item there exists a 0-object with $0\rightarrowtail A$, and
\item for every commuting diagram of the form
\[
\begin{tikzpicture}[xscale=1.3,yscale=1.4]
\node (cp) at (0,0) {$C'$};
\node (c) at (0,1) {$C$};
\node (a) at (1,1) {$A$};
\node (ap) at (1,0) {$A'$};
\node (b) at (2,1) {$B$};
\node (bp) at (2,0) {$B'$,};
\draw[->] (a)--(c);
\draw[->] (ap)--(cp);
\draw[>->] (a)--(b);
\draw[>->] (ap)--(bp);
\foreach \x\y in {c/cp, a/ap, b/bp}{
  \draw[->] (\x) to node[right=6pt,pos=.2,rotate=270] {$\sim$} (\y);
}
\end{tikzpicture}
\] 
we have $B\cup_A C\wequiv B'\cup_{A'} C'$.
\end{enumerate}
\end{defn}

\begin{defn}\index{Waldhausen category}\index{categroy!Waldhausen}
A \emph{Waldhausen category} is a category with cofibrations and weak equivalences. 
\end{defn}

We write $K_0(\mathcal C)$, for $\mathcal C$ Waldhausen, to denote the abelian group with generators $[C]$ for every $C\in \obj (\mathcal C)$, and relations $[C]=[C']$ for every $C\wequiv C'$. For every \emph{cofiber sequence} $B\cofib C \twoheadrightarrow B/C$, we also have $[C] = [B]+[B/C]$. From the properties of a Waldhausen category, we immediately have that 
\begin{itemize}
\item $[0] = 0$,
\item $[C]=0$ if $C\wequiv 0$, 
\item $[B\cup_A C] = [B]+[C]-[A]$.
\end{itemize}

Recall that a functor is \emph{exact} if it preserves the zero object, cofibrations, weak equivalences and pushouts.

\begin{defn}
A \emph{Waldhausen subcategory} $\mathcal A$ of a Waldhausen category $\mathcal C$ is such that
\begin{itemize}
\item $\mathcal A\subseteq \mathcal C$ is exact,
\item cofibrations in $\mathcal A$ are maps in $\mathcal A$ which are cofibrations in $\mathcal C$ whose cokernel lies in $\mathcal A$, 
\item weak equivalences in $\mathcal A$ are weak equivalences in $\mathcal C$.
\end{itemize}
\end{defn}

\noindent
Now define a sequence of Waldhausen categories, for $\mathcal C$ a category with cofibrations, in the following way:
\begin{center}
\begin{tabular}{r c l}
$S_0\mathcal C$ & $=$ & 0, \\
$S_1\mathcal C$ & $=$ & $\mathcal C$, objects $0\cofib A$, \\
$S_2\mathcal C$ & & has objects cofiber sequences $E:A_1\cofib A_2\twoheadrightarrow A_{12}$ and morphisms\\
& & ${}$\hspace{1.5cm} \begin{tikzpicture}[xscale=1.4,yscale=1.2]
\node (ep) at (-.6,0) {$E'$\vphantom{$A_1'$}};
\node (e) at (-.6,1) {$E$};
\node (cp) at (0,0) {$A_1'$};
\node (c) at (0,1) {$A_1$};
\node (a) at (1,1) {$A_2$};
\node (ap) at (1,0) {$A_2'$};
\node (b) at (2,1) {$A_{12}$};
\node (bp) at (2,0) {$A_{12}'$.};
\draw[->] (e) to node[right] {$f$} (ep);
\draw[<-<] (a)--(c);
\draw[<-<] (ap)--(cp);
\draw[->>] (a)--(b);
\draw[->>] (ap)--(bp);
\foreach \x\y\z in {c/cp/1, a/ap/2, b/bp/12}{
  \draw[->] (\x) to node[right] {$u_{\z}$} (\y);
}
\node at ($(e.east)!.5!(c.west)$) {:};
\node at ($(ep.east)!.5!(cp.west)$) {:};
\end{tikzpicture}\\
& & A map $f\in S_2\mathcal C$ is a cofibration if $u_1,u_{12}$ are cofibrations and $A_1'\cup_{A_1} A_2\to A_2'$ is a cofibration. \\ & & A map $f$ is a weak equivalence if all of $u_1,u_2,u_{12}$ are weak equivalences.\\[5pt]
$S_n \mathcal C$ & & has objects $A_\bullet: 0 =A_0 \cofib A_1 \cofib \cdots \cofib A_n$ together with a choice of quotients $A_i/A_{i-1}$. \\ & & The following diagram $(\Delta)$ commutes:\\
& & ${}$ \hspace{1.5cm} \begin{tikzpicture}[xscale=1.2,yscale=1.1]
\node (a1) at (0,0) {$A_1$};
\node (a2) at (1,0) {$A_2$};
\node (a3) at (2,0) {$A_3$};
\node (a4) at (3,0) {$\cdots$};
\node (an) at (4,0) {$A_n$};
\foreach \x\y\z in {1/1/12, 2/1/13, 2/2/23, 4/1/1n, 4/2/2n, 4/4/nn}{
  \node (a\z) at (\x,\y) {$A_{\z}$};
} 
\node (a14) at (3,1) {$\cdots$};
\node (a24) at (3,2) {$\cdots$};
\node (an4) at (3,4) {$\cdots$};
\node (a3n) at (4,3) {$\vdots$};
\foreach \x\y in {a1/a2, a2/a3, a3/a4, a4/an, a12/a13, a13/a14, a14/a1n, a23/a24, a24/a2n, an4/ann}{
  \draw[>->] (\x)--(\y);
}
\foreach \x\y in {a2/a12, a3/a13, a13/a23, an/a1n, a1n/a2n, a2n/a3n, a3n/ann}{
  \draw[->>] (\x)--(\y);
}
\end{tikzpicture}\\
& & Morphisms of $S_n\mathcal C$ are natural tranformations of cofiber sequences.
\end{tabular}
\end{center}

Note that $S_n\mathcal C$ is Waldhausen.

\subsection{Simplicial Waldhausen categories}

The payoff from this long construction is that we get a \emph{simplicial} Waldhausen category. We get $\partial_0:S_n\mathcal C\to S_{n-1}\mathcal C$ by deleting the bottom row in $(\Delta)$. That is, 
\[
\partial_0(A_\bullet) = \left(0 = A_{11} \cofib A_{12} \cofib A_{13}\cofib \cdots \cofib A_{1n}\right)
\]
together with a choice of quotients. We also have $\partial_0(A_\bullet)_{ij} = A_{i+1,j+1}$. Note that $\partial_0$ is exact. For $0< i\leqslant n$, we have $\partial_i:S_n\mathcal C \to S_{n-1}\mathcal C$ by omitting $A_{i*}$ and the column containing $A_i$ in $(\Delta)$. Dually, we get $S_i:S_n\mathcal C\to S_{n+1}\mathcal C$ by duplicating the column that contains $A_i$ and reindexing.

\begin{rem}
This shows that $S_\bullet\mathcal C$ is a simplicial Waldhausen category and the full subcategory $wS_\bullet \mathcal C$ of weak equivalences is a simplicial Waldhausen subcategory of $S_\bullet\mathcal C$.
\end{rem}

The geometric realization $|wS_\bullet \mathcal C|$ is a connected simplicial space. It may be shown that $\pi_1|wS_\bullet \mathcal C| = K_0(\mathcal C)$, and the $\Sigma$-$\Omega$ (suspension-loop space) adjunction gives that $K_i(\mathcal C) = \pi_{i+1}|wS_\bullet\mathcal C|$.




\clearpage
\nocite{*}
\addcontentsline{toc}{section}{References}
\printbibliography

\clearpage
\section*{Index of notation}
\addcontentsline{toc}{section}{Index of notation}
\renewcommand\arraystretch{1.2}
\begin{longtable}{l l l}
$\Delta$ & category of finite ordinals & \pageref{not-delta} \\
$\sSet$ & category of simplicial sets & \pageref{not-delta} \\
$\Delta^n$ & standard $n$-simplex & \pageref{not-deltan} \\
$N(C)$ & nerve of a category $C$ & \pageref{not-nerve} \\
$\Lambda^n_i$ & $i$th $n$-horn & \pageref{not-horn} \\
$\Ho(\mathcal C)$ & homotopy category of a category $\mathcal C$ & \pageref{not-homcat}, \pageref{not-hocat}\\
$QC(X)$ & category of quasi-coherent sheaves on $X$ & \pageref{not-qcs} \\
$\mathcal C_{p/}$, $\mathcal C_{/p}$ & slice category, (co)cone of diagrams over a category $\mathcal C$ & \pageref{not-slice} \\
$\mathcal C\star \mathcal D$ & join of categories $\mathcal C$ and $\mathcal D$ & \pageref{not-join} \\
$\hookrightarrow$, $\twoheadrightarrow$, $\xrightarrow\sim$ & cofibration, fibration, and weak equivalence in a model category & \pageref{not-modelcat} \\
$A_C$, $A_F$, $A_{CF}$ & cofibrant, fibrant, and bifibrant replacements of an object $A$ & \pageref{not-fibcofib} \\
$A\wedge I$, $A^I$ & cylinder object and path object of $A$ in a model category & \pageref{not-smashpath} \\
$\Shv(X)$, $\Shv_{LC}(X)$ & category of (locally constant) sheaves on a topological space $X$ & \pageref{not-sheaves} \\
$\Shv^A(X)$ & category of $A$-constructible sheaves, for $A$ a poset & \pageref{not-constructiblesheaf} \\
$\Sing^A(X)$ & category of exit paths on an $A$-stratified space $X$ & \pageref{not-exitpath} \\
$\mathcal C_d$ & fiber of a category $\mathcal C$ over a functor $p:\mathcal C\to \mathcal D\owns d$ & \pageref{catfiber} \\
$\alpha!$ & functor induced by a Grothendieck opfibration & \pageref{abang} \\
$\Pr^L$ & category of presentable $\infty$-categories with colimit preserving functors & \pageref{prl} \\
$\Perf(A)$ & category of perfect $A$-modules & \pageref{perfectmodule} \\
$E$ & prespectrum or spectrum & \pageref{spectrum} \\
$\mathcal C^\Omega$, $\mathcal C^\Sigma$ & category of exact, coexact triangles in $\mathcal C$ & \pageref{exacttriangles} \\
$\vee$, $\wedge$ & join and meet & \pageref{meetjoin} \\
$A\cofib B$, $A \wequiv B$ & cofibration, weak equivalence from $A$ to $B$ & \pageref{arrows}
\end{longtable}
\renewcommand\arraystretch{1}

\clearpage
\addcontentsline{toc}{section}{Index}
\printindex

\end{document}