nLab
total category

Contents

Idea

A total category is a category with a well-behaved Yoneda embedding endowing the category with very good completeness and cocompleteness properties but still admitting most types of categories occurring “in practice”.

Definition

Definition

A locally small category CC is total if its Yoneda embedding Y:C[C op,Set]Y \;\colon \;C\longrightarrow [C^{op},Set] has a left adjoint LL.

If the opposite category C opC^{op} is total, CC is called cototal.

Remark

The definition above requires some set-theoretic assumption to ensure that the functor category [C op,Set][C^{op},Set] exists, but it can be rephrased to say that the colimit of Id C:CCId_C:C\to C weighted by WW exists, for any W:C opSetW:C^{op}\to Set. (This still involves quantification over large objects, however, so some foundational care is needed.) This version has an evident generalization to enriched categories.

Remark

Since the Yoneda embedding is a full and faithful functor, a total category CC induces an idempotent monad YLY \circ L on its category of presheaves, hence a modality. One says that CC is a totally distributive category if this modality is itself the right adjoint of an adjoint modality.

Remark

The (LY)(L \dashv Y)-adjunction of a total category is closely related to the (𝒪Spec)(\mathcal{O} \dashv Spec)-adjunction discussed at Isbell duality and at function algebras on ∞-stacks. In that context the LYL Y-modality deserves to be called the affine modality.

Properties

  • Total categories satisfy a very satisfactory adjoint functor theorem: any colimit-preserving functor from a total category to a locally small category has a right adjoint.

  • Although the definition refers explicitly only to colimits, every total category is also complete, i.e. has all small limits. It also has some large limits. In fact, it has “all possible” large limits that a locally small category can have: if F:DCF\colon D\to C is a functor such that lim dHom C(X,Fd)lim_d Hom_C(X,F d) is a small set for all XCX\in C, then FF has a limit.

  • A total category 𝒞\mathcal{C} is cartesian closed iff LL preserves binary products (cf. Wood 1982, thm. 9).

Examples

Proposition

Any cocomplete and epi-cocomplete category with a generator is total. More generally, any cocomplete and EE-complete category with an EE-generator is total, for a suitable class EE.

See (Day), theorem 1, for a proof. This includes:

Also, totality lifts along solid functors; that is, if the codomain of a solid functor is total, then so is its domain. See (Tholen) for a proof. This implies that the following types of categories are total:

For example

  • any category which is monadic over Set

  • any category admitting a topological functor to Set

  • The category Grp of groups as a category monadic over SetSet is total, but it is not cototal; see below.

  • The category of topological groups is total as well since it is topological over the total category Grp.

  • If CC is total and JJ is small, then C JC^J is total, morally because it is a reflective subcategory of Set C op×JSet^{C^{op} \times J}; see section 6 of Kelly.

Thus, “most naturally-occurring” cocomplete categories are in fact total.

In practice, i.e., in naturally occurring concrete cases, cototality is more rare. For example, it is frequently not the case that categories that are monadic over SetSet are cototal. This is well-illustrated by the following two examples:

  • The category of groups Grp is not cototal; if it were, then any continuous functor GrpSetGrp \to Set would be representable. To see this is not the case, it suffices to produce a class of simple groups G αG_\alpha of unbounded cardinality (for example, for any infinite set XX, the alternating group Alt(X)Alt(X), consisting of permutations of finite support that are even, is simple and of cardinality equal to that of XX). For any group GG, the hom-set hom(G α,G)\hom(G_\alpha, G) consists of a single element (the trivial homomorphism) as soon as the cardinality of G αG_\alpha exceeds that of GG. Thus the class-indexed product αhom(G α,G)\prod_\alpha \hom(G_\alpha, G) is bounded in size, and defines a continuous functor F= αhom(G α,):GrpSetF = \prod_\alpha \hom(G_\alpha, -): Grp \to Set. But it is clear this functor is not representable; e.g., for any group GG, one can find G αG_\alpha such that F(G α)F(G_\alpha) is much larger in size than hom(G,G α)\hom(G, G_\alpha). This example is given in Wood 1982.

  • By a similar construction, the category of commutative rings is not cototal. For each infinite cardinal α\alpha, choose a field F αF_\alpha of size α\alpha, e.g., an algebraically closed field over \mathbb{Q} of transcendence degree α\alpha. Put A α=×F αA_\alpha = \mathbb{Z} \times F_\alpha. Then, for any commutative ring RR, there is exactly one homomorphism A αRA_\alpha \to R as soon as α\alpha exceeds the cardinality of RR. Then one argues that αhom(A α,):CRingSet\prod_\alpha \hom(A_\alpha, -): CRing \to Set is continuous but not representable.

But cototal categories do occur:

References

  • Ross Street, Bob Walters, Yoneda structures on 2-category, (contains the original definition of total categories)

  • Max Kelly, A survey of totality for enriched and ordinary categories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 2 (1986), p. 109-132, numdam

  • Brian Day, Further criteria for totality, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 28 no. 1 (1987), p. 77-78, numdam
  • Richard Garner, Topological=Total , arXiv:1310.0903 (2013). (abstract)

  • Ross Street, The family approach to total cocompleteness and toposes , Trans. A. M. S. 284 (1978) pp.355-369.

  • Richard J. Wood, Some remarks on total categories, J. Algebra 75_:2, 1982, 538–545 doi

Last revised on March 26, 2018 at 08:23:27. See the history of this page for a list of all contributions to it.