nLab
clique

Definition

A 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 collection of objects) and a functor $A:T\to C$.

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

We can form a category $\mathrm{Clique}(C)$ 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 $m:(T_0,A_0)\to (T_1,A_1)$ and $n:(T_1,A_1)\to (T_2,A_2)$, and $(t_0,t_2)\in \mathrm{Ob}(T_0\times T_2)$, 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 $t_1\in \mathrm{Ob}(T_1)$, and there is at least one such choice. Accordingly, we can take this to give a well-defined component $(nm{)}_{(t_0,t_2)}$, 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 $(t,t\prime )\in \mathrm{Ob}(T\times T)$ is the value of $A$ on the unique morphism from $t$ to $t\prime$ in $T$.

Applications

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 $a\overleftarrow{p}c\overrightarrow{q}b$, 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.

Cliques and anafunctors

There is an obvious anafunctor from $\mathrm{Clique}(C)$ into $C$, through which every other anafunctor into $C$ factors in an essentially unique way into a genuine functor. This induces for $\mathrm{Clique}(-)$ the structure of a (2-)monad on $\mathrm{Str}\mathrm{Cat}$ (the (2-)category of “genuine” functors between categories), such that the Kleisli category for this monad will be ${\mathrm{Cat}}_{\mathrm{ana}}$ (the (2-)category of anafunctors between categories). This monad can also be described more explicitly; in particular the unit (a “genuine” functor) $C\to \mathrm{Clique}(C)$ takes each object $c\in C$ to the corresponding clique $c:1\to C$ 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 define anafunctors, taking an anafunctor from $C$ into $D$ to simply be a genuine functor from $C$ into $\mathrm{Clique}(D)$. (With composition of these defined in a straightforward way, and natural transformations between these being simply natural transformations of the corresponding genuine functors into $\mathrm{Clique}(D)$). Accordingly, $\mathrm{Clique}(-)$ is itself the same as ${\mathrm{Cat}}_{\mathrm{ana}}(1,-)$, and this can also be taken as a definition of a clique (hence the alternate name anaobject).

Monoidal strictifications

Unsurprisingly, cliques provide a useful technical device for describing strictifications of monoidal categories.

It is relevant first to recall the original form of Mac Lane’s 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:

1. We may form a monoidal category $\mathrm{Oper}(M)$ whose objects are functors

   $$F:M^j\to M$$F: M^j \to M

   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 $\mathrm{Oper}(M)$ is the composite

   $$M^{j+k}\cong M^j\times M^k\stackrel{F\times G}{\to }M\times M\stackrel{\otimes }{\to }M$$M^{j+k} \cong M^j \times M^k \stackrel{F \times G}{\to} M \times M \stackrel{\otimes}{\to} M

   and the rest of the monoidal structure on $\mathrm{Oper}(M)$ is inherited from the monoidal structure on $M$.

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

   $$\kappa :F[1]\to \mathrm{Oper}(M)$$\kappa: F[1] \to Oper(M)

   uniquely determined as the one which sends the generator $1$ of $F[1]$ to ${\mathrm{Id}}_M$. On each connected component $C_n$ of $F[1]$, this restricts to a functor

   $$C_n\stackrel{\kappa \mid }{\to }\mathrm{Cat}(M^n,M)$$C_n \stackrel{\kappa|}{\to} Cat(M^n, M)

3. Then, for each $n$-tuple of objects $(x_1,\dots ,x_n)$ of objects of $M$, there is an associated clique ${\kappa }_{x_1,\dots ,x_n}$ in $M$:

   $$C_n\stackrel{\kappa \mid }{\to }\mathrm{Cat}(M^n,M)\stackrel{{\mathrm{eval}}_{(x_1,\dots ,x_n)}}{\to }M$$C_n \stackrel{\kappa|}{\to} Cat(M^n, M) \stackrel{eval_{(x_1, \ldots, x_n)}}{\to} M

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

   $$(x_1,\dots ,x_m)\to (y_1,\dots ,y_n)$$(x_1, \ldots, x_m) \to (y_1, \ldots, y_n)

   is by definition a clique morphism ${\kappa }_{x_1,\dots ,x_m}\to {\kappa }_{y_1,\dots ,y_n}$. There is an evident strict monoidal category structure on $M^{\mathrm{st}}$ which at the object level is just concatenation of tuples.

It is straightforward to check that the natural inclusion

$$i:M\to M^{\mathrm{st}},$$i: M \to M^{st},

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

$$(x_1,x_2,\dots ,x_n)\to i(\mathrm{Bracketing}(x_1\otimes \dots \otimes x_n))$$(x_1, x_2, \ldots, x_n) \to i(Bracketing(x_1 \otimes \ldots \otimes x_n))

for every choice of bracketing the tensor product on the right in $M$ (possibly with units thrown in).

Etymology and relation to graph theory

There is a notion of clique in an undirected (simple) graph familiar to graph-theorists: a clique $C$ in a graph $G$ is a subgraph which is complete? as a graph, i.e., one for which any two distinct vertices are connected by an edge. Thus, a clique having $n$ vertices is isomorphic to an inclusion of a $K_n$.

A reasonable analogue for quivers (the category theorists' directed graphs) might be a subgraph $C$ which is 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.