Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts





A clique of a category CC is a functor TCT \to C from a (-2)-groupoid TT, or equivalently an anafunctor to CC from the trivial category.

So this is a pair of a category TT which is weakly equivalent to 11 (i.e., TT is the indiscrete category on an inhabited collection of objects) and a functor A:TCA\colon T \rightarrow C.

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

We can form a category Clique(C)Clique(C) whose objects are cliques of CC, and whose morphisms and compositions are given as follows: Given two such cliques (T 0,A 0)(T_0, A_0) and (T 1,A 1)(T_1, A_1) in CC, say that a morphism between them is a natural transformation from T 0×T 1πT 0A 0CT_0 \times T_1 \stackrel{\pi}{\to} T_0 \stackrel{A_0}{\to} C to T 0×T 1πT 1A 1CT_0 \times T_1 \stackrel{\pi}{\to} T_1 \stackrel{A_1}{\to} C, where the π\pi are the appropriate projections. Given such morphisms m:(T 0,A 0)(T 1,A 1)m : (T_0, A_0) \rightarrow (T_1, A_1) and n:(T 1,A 1)(T 2,A 2)n : (T_1, A_1) \rightarrow (T_2, A_2), and (t 0,t 2)Ob(T 0×T 2)(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)n_{(t_1, t_2)} m_{(t_0, t_1)} of corresponding components has the same value no matter what the choice of t 1Ob(T 1)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 (nm) (t 0,t 2)(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)(T, A) to be the natural transformation whose component on (t,t)Ob(T×T)(t, t') \in Ob(T \times T) is the value of AA on the unique morphism from tt to tt' in TT.


Objects with universal properties

Many universal properties that are commonly considered as defining “an object” actually define a clique. For example, given two objects aa and bb of a category CC, their cartesian product can be considered as the clique TCT\to C, where TT is the indiscrete category whose objects are product diagrams apcqba \overset{\leftarrow}{p} c \overset{\to}{q} b, and where the functor TCT\to C sends each such diagram to the object cc and each morphism to the unique comparison isomorphism between two cartesian products. Note that unlike “the product” of aa and bb 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 Clique(C)Clique(C) into CC, through which every other anafunctor into CC factors in an essentially unique way into a genuine functor. This induces for Clique()Clique(-) the structure of a (2-)monad on StrCatStr Cat (the (2-)category of “genuine” functors between categories), such that the Kleisli category for this monad will be Cat anaCat_{ana} (the (2-)category of anafunctors between categories). This monad can also be described more explicitly; in particular the unit (a “genuine” functor) CClique(C)C\to Clique(C) takes each object cCc\in C to the corresponding clique c:1Cc\colon 1\to C defined on the domain 11. 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 CC into DD to simply be a genuine functor from CC into Clique(D)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)Clique(D)). Accordingly, Clique()Clique(-) is itself the same as Cat ana(1,)Cat_{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]F[1], is monoidally equivalent to the discrete monoidal category (,+,0)(\mathbb{N}, +, 0). Thus each connected component C nC_n of F[1]F[1] is an indiscrete category whose objects are the possible nn-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 MM is by considering cliques in MM where the domains are the C nC_n and the functors model associativity and unit constraints, in the following precise sense:

  1. We may form a monoidal category Oper(M)Oper(M) whose objects are functors

    F:M jMF: M^j \to M

    and whose morphisms are natural transformations between such functors. The tensor product of F:M jMF: M^j \to M and G:M kMG: M^k \to M in Oper(M)Oper(M) is the composite

    M j+kM j×M kF×GM×MMM^{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 Oper(M)Oper(M) is inherited from the monoidal structure on MM.

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

    κ:F[1]Oper(M)\kappa: F[1] \to Oper(M)

    uniquely determined as the one which sends the generator 11 of F[1]F[1] to Id MId_M. On each connected component C nC_n of F[1]F[1], this restricts to a functor

    C nκ|Cat(M n,M)C_n \stackrel{\kappa|}{\to} Cat(M^n, M)
  3. Then, for each nn-tuple of objects (x 1,,x n)(x_1, \ldots, x_n) of objects of MM, there is an associated clique κ x 1,,x n\kappa_{x_1, \ldots, x_n} in MM:

    C nκ|Cat(M n,M)eval (x 1,,x n)MC_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 stM^{st} are nn-tuples (x 1,,x n)(x_1, \ldots, x_n) of objects of MM. A morphism

    (x 1,,x m)(y 1,,y n)(x_1, \ldots, x_m) \to (y_1, \ldots, y_n)

    is by definition a clique morphism κ x 1,,x mκ y 1,,y n\kappa_{x_1, \ldots, x_m} \to \kappa_{y_1, \ldots, y_n}. There is an evident strict monoidal category structure on M stM^{st} which at the object level is just concatenation of tuples.

It is straightforward to check that the natural inclusion

i:MM 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,,x n)i(Bracketing(x 1x 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 MM (possibly with units thrown in).

In graph theory

There is a notion of clique in an undirected simple graph familiar to graph-theorists: a clique in a graph GG is a subset of vertices CV(G)C \subseteq V(G) such that any two distinct vertices x,yCx,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 nn) in GG to be a graph homomorphism C:K nGC : K_n \to G from the complete graph? on nn 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 CC of GG which is indiscrete: there is exactly one edge in CC from xx to yy for any vertices xx, yy of CC.

The categorical notion of clique is one step removed from that: a clique in a category CC is a functor i:KCi: K \to C where the underlying graph of KK 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.

Revised on October 20, 2015 07:40:38 by Noam Zeilberger (