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\begin{document}

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\section*{clique}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]=--

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{objects_with_universal_properties}{Objects with universal properties}\dotfill \pageref*{objects_with_universal_properties} \linebreak
\noindent\hyperlink{cliques_and_anafunctors}{Cliques and anafunctors}\dotfill \pageref*{cliques_and_anafunctors} \linebreak
\noindent\hyperlink{monoidal_strictifications}{Monoidal strictifications}\dotfill \pageref*{monoidal_strictifications} \linebreak
\noindent\hyperlink{in_graph_theory}{In graph theory}\dotfill \pageref*{in_graph_theory} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{clique} of a category $C$ is a [[functor]] $T \to C$ from a [[(-2)-groupoid]] $T$, or equivalently an [[anafunctor]] to $C$ from the [[trivial category]].

So this is a pair of a category $T$ which is weakly equivalent to $1$ (i.e., $T$ is the [[indiscrete category]] on an [[inhabited set|inhabited collection]] of [[object]]s) and a [[functor]] $A\colon T \rightarrow C$.

A clique is also sometimes called an \textbf{anaobject}, since an [[object]] of $C$ is a [[functor]] (not anafunctor) to $C$ from the trivial category.

We can form a category \textbf{$Clique(C)$} whose objects are cliques of $C$, and whose morphisms and compositions are given as follows: Given two such cliques $(T_0, A_0)$ and $(T_1, A_1)$ in $C$, say that a morphism between them is a natural transformation from $T_0 \times T_1 \stackrel{\pi}{\to} T_0 \stackrel{A_0}{\to} C$ to $T_0 \times T_1 \stackrel{\pi}{\to} T_1 \stackrel{A_1}{\to} C$, where the $\pi$ are the appropriate projections. Given such morphisms $m : (T_0, A_0) \rightarrow (T_1, A_1)$ and $n : (T_1, A_1) \rightarrow (T_2, A_2)$, and $(t_0, t_2) \in Ob(T_0 \times T_2)$, note that the composite $n_{(t_1, t_2)} m_{(t_0, t_1)}$ of corresponding components has the same value no matter what the choice of $t_1 \in Ob(T_1)$, and there is at least one such choice. Accordingly, we can take this to give a well-defined component $(n m)_{(t_0, t_2)}$, thus defining binary composition of morphisms of cliques. Similarly, we can take the identity on a clique $(T, A)$ to be the natural transformation whose component on $(t, t') \in Ob(T \times T)$ is the value of $A$ on the unique morphism from $t$ to $t'$ in $T$.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{objects_with_universal_properties}{}\subsubsection*{{Objects with universal properties}}\label{objects_with_universal_properties}

Many [[universal properties]] that are commonly considered as defining ``an object'' actually define a clique. For example, given two objects $a$ and $b$ of a category $C$, their [[cartesian product]] can be considered as the clique $T\to C$, where $T$ is the indiscrete category whose objects are product diagrams $a \overset{\leftarrow}{p} c \overset{\to}{q} b$, and where the functor $T\to C$ sends each such diagram to the object $c$ and each morphism to the unique comparison isomorphism between two cartesian products. Note that unlike ``the product'' of $a$ and $b$ considered as a single object, this clique is defined without making any arbitrary choices. This of course is the same philosophy which leads to [[anafunctors]], and so cliques are closely related to anafunctors.

\hypertarget{cliques_and_anafunctors}{}\subsubsection*{{Cliques and anafunctors}}\label{cliques_and_anafunctors}

There is an obvious [[anafunctor]] from $Clique(C)$ into $C$, through which every other anafunctor into $C$ factors in an essentially unique way into a genuine functor. This induces for $Clique(-)$ the structure of a (2-)monad on $Str Cat$ (the (2-)category of ``genuine'' functors between categories), such that the [[Kleisli category]] for this monad will be $Cat_{ana}$ (the (2-)category of anafunctors between categories). This monad can also be described more explicitly; in particular the unit (a ``genuine'' functor) $C\to Clique(C)$ takes each object $c\in C$ to the corresponding clique $c\colon 1\to C$ defined on the domain $1$. Note that this functor is a [[weak equivalence]], i.e. fully faithful and essentially surjective on objects, but not a strong equivalence unless one assumes the [[axiom of choice]].

In particular, we can use cliques to \emph{define} anafunctors, taking an anafunctor from $C$ into $D$ to simply be a genuine functor from $C$ into $Clique(D)$. (With composition of these defined in a straightforward way, and natural transformations between these being simply natural transformations of the corresponding genuine functors into $Clique(D)$). Accordingly, $Clique(-)$ is itself the same as $Cat_{ana}(1, -)$, and this can also be taken as a definition of a clique (hence the alternate name \emph{anaobject}).

\hypertarget{monoidal_strictifications}{}\subsubsection*{{Monoidal strictifications}}\label{monoidal_strictifications}

Unsurprisingly, cliques provide a useful technical device for describing strictifications of [[monoidal category|monoidal categories]].

It is relevant first to recall the original form of Mac Lane's [[coherence theorems|coherence theorem]]: the free monoidal category on one generator, $F[1]$, is monoidally equivalent to the discrete monoidal category $(\mathbb{N}, +, 0)$. Thus each connected component $C_n$ of $F[1]$ is an indiscrete category whose objects are the possible $n$-fold tensor products of the generator, possibly with instances of the unit object folded in; the indiscreteness says that ``all diagrams built from associativity and unit constraints commute''.

One canonical way to strictify a monoidal category $M$ is by considering cliques in $M$ where the domains are the $C_n$ and the functors model associativity and unit constraints, in the following precise sense:

\begin{enumerate}%
\item We may form a monoidal category $Oper(M)$ whose objects are functors

\begin{displaymath}
F: M^j \to M
\end{displaymath}
and whose morphisms are natural transformations between such functors. The tensor product of $F: M^j \to M$ and $G: M^k \to M$ in $Oper(M)$ is the composite

\begin{displaymath}
M^{j+k} \cong M^j \times M^k \stackrel{F \times G}{\to} M \times M \stackrel{\otimes}{\to} M
\end{displaymath}
and the rest of the monoidal structure on $Oper(M)$ is inherited from the monoidal structure on $M$.


\item By freeness of $F[1]$, we have a (strict) monoidal functor

\begin{displaymath}
\kappa: F[1] \to Oper(M)
\end{displaymath}
uniquely determined as the one which sends the generator $1$ of $F[1]$ to $Id_M$. On each connected component $C_n$ of $F[1]$, this restricts to a functor

\begin{displaymath}
C_n \stackrel{\kappa|}{\to} Cat(M^n, M)
\end{displaymath}

\item Then, for each $n$-tuple of objects $(x_1, \ldots, x_n)$ of objects of $M$, there is an associated clique $\kappa_{x_1, \ldots, x_n}$ in $M$:

\begin{displaymath}
C_n \stackrel{\kappa|}{\to} Cat(M^n, M) \stackrel{eval_{(x_1, \ldots, x_n)}}{\to} M
\end{displaymath}

\item Finally, the objects of the strictification $M^{st}$ are $n$-tuples $(x_1, \ldots, x_n)$ of objects of $M$. A morphism

\begin{displaymath}
(x_1, \ldots, x_m) \to (y_1, \ldots, y_n)
\end{displaymath}
is by definition a clique morphism $\kappa_{x_1, \ldots, x_m} \to \kappa_{y_1, \ldots, y_n}$. There is an evident strict monoidal category structure on $M^{st}$ which at the object level is just concatenation of tuples.



\end{enumerate}
It is straightforward to check that the natural inclusion

\begin{displaymath}
i: M \to M^{st},
\end{displaymath}
which interprets each object as a 1-tuple and each morphism as an evident clique morphism, is a monoidal equivalence. The essential idea is that there is a canonical clique isomorphism

\begin{displaymath}
(x_1, x_2, \ldots, x_n) \to i(Bracketing(x_1 \otimes \ldots \otimes x_n))
\end{displaymath}
for every choice of bracketing the tensor product on the right in $M$ (possibly with units thrown in).

\hypertarget{in_graph_theory}{}\subsection*{{In graph theory}}\label{in_graph_theory}

There is a notion of clique in an undirected simple [[graph]] familiar to graph-theorists: a \emph{clique} in a graph $G$ is a subset of vertices $C \subseteq V(G)$ such that any two distinct vertices $x,y \in C$ are connected by an edge. This definition is specialized to simple graphs, however, and a more general definition that works for arbitrary undirected graphs (possibly containing loops and multiple edges) takes a clique (of size $n$) in $G$ to be a graph homomorphism $C : K_n \to G$ from the [[complete graph]] on $n$ vertices. Indeed, this latter definition could also be taken as a reasonable notion of clique in any undirected graph/[[quiver]]. Equivalently, a clique in this sense is a subgraph $C$ of $G$ which is \emph{indiscrete}: there is exactly one edge in $C$ from $x$ to $y$ for any vertices $x$, $y$ of $C$.

The categorical notion of clique is one step removed from that: a clique in a category $C$ is a functor $i: K \to C$ where the underlying graph of $K$ is indiscrete. The generic ``picture'' of a clique in a category is reminiscent of (and no doubt the etymology derives from) the graph-theoretic notion, even if the notions are technically distinct.

[[!redirects clique]] [[!redirects cliques]]



\end{document}