We can form a category whose objects are cliques of , and whose morphisms and compositions are given as follows: Given two such cliques and in , say that a morphism between them is a natural transformation from to , where the are the appropriate projections. Given such morphisms and , and , note that the composite of corresponding components has the same value no matter what the choice of , and there is at least one such choice. Accordingly, we can take this to give a well-defined component , thus defining binary composition of morphisms of cliques. Similarly, we can take the identity on a clique to be the natural transformation whose component on is the value of on the unique morphism from to in .
Many universal properties that are commonly considered as defining “an object” actually define a clique. For example, given two objects and of a category , their cartesian product can be considered as the clique , where is the indiscrete category whose objects are product diagrams , and where the functor sends each such diagram to the object and each morphism to the unique comparison isomorphism between two cartesian products. Note that unlike “the product” of and 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.
There is an obvious anafunctor from into , through which every other anafunctor into factors in an essentially unique way into a genuine functor. This induces for the structure of a (2-)monad on (the (2-)category of “genuine” functors between categories), such that the Kleisli category for this monad will be (the (2-)category of anafunctors between categories). This monad can also be described more explicitly; in particular the unit (a “genuine” functor) takes each object to the corresponding clique defined on the domain . 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 into to simply be a genuine functor from into . (With composition of these defined in a straightforward way, and natural transformations between these being simply natural transformations of the corresponding genuine functors into ). Accordingly, is itself the same as , and this can also be taken as a definition of a clique (hence the alternate name anaobject).
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, , is monoidally equivalent to the discrete monoidal category . Thus each connected component of is an indiscrete category whose objects are the possible -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 is by considering cliques in where the domains are the and the functors model associativity and unit constraints, in the following precise sense:
We may form a monoidal category whose objects are functors
and whose morphisms are natural transformations between such functors. The tensor product of and in is the composite
and the rest of the monoidal structure on is inherited from the monoidal structure on .
By freeness of , we have a (strict) monoidal functor
uniquely determined as the one which sends the generator of to . On each connected component of , this restricts to a functor
Then, for each -tuple of objects of objects of , there is an associated clique in :
Finally, the objects of the strictification are -tuples of objects of . A morphism
is by definition a clique morphism . There is an evident strict monoidal category structure on which at the object level is just concatenation of tuples.
It is straightforward to check that the natural inclusion
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
for every choice of bracketing the tensor product on the right in (possibly with units thrown in).
There is a notion of clique in an undirected simple graph familiar to graph-theorists: a clique in a graph is a subset of vertices such that any two distinct vertices 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 ) in to be a graph homomorphism from the complete graph? on 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 of which is indiscrete: there is exactly one edge in from to for any vertices , of .
The categorical notion of clique is one step removed from that: a clique in a category is a functor where the underlying graph of 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.