nLab total category

Idea

A total category (also called a totally cocomplete category, or an absolutely cocomplete category) is a category whose Yoneda embedding has a left adjoint. This totality condition implies various strong cocompleteness and completeness properties that are especially convenient in the context of an adjoint functor theorem.

In the words of Street, “Total categories are precisely the categories, algebraic and topological, at which traditional category theory was aimed”. In other words, categories like Set, Ab, Grp, Top, TopGrp: traditional large categories of “mathematical structures”. Roughly speaking, for the various species of structure in the zoology of Bourbaki (structures of algebraic type, topological type, order type, or some hybrid), if we consider species of a given structure type, and homomorphisms preserving that type of structure, as forming a category, then those sorts of categories have a strong tendency to be total. More precise theorems to this effect are given below.

Definition

Definition

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

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

Remark

The definition above requires some set-theoretic assumption to ensure that the functor category $[C^{op},Set]$ exists, but it can be rephrased to say that the colimit of $Id_C:C\to C$ weighted by $W$ exists, for any $W:C^{op}\to Set$. (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 $C$ induces an idempotent monad $Y \circ L$ on its category of presheaves, hence a modality. One says that $C$ is a totally distributive category if this modality is itself the right adjoint of an adjoint modality.

Remark

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

Properties

• Every total category has all small colimits.

• 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. (See Wood 1982, Thm. 1 for a clear statement of this fact and StreetWalters, p. 372 for a proof).

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

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

• Recall that every functor $f : A \to X$ factors via the comprehensive factorisation as a final functor $e$ follows by a discrete fibration $m$. A locally small category $X$ is total if and only if for every such functor $f$, such that the discrete fibration $m$ has small fibres, has a colimit. (In fact, this definition makes sense even when $X$ is not locally small.)

Examples

Proposition

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

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

• locally presentable categories, hence in particular Grothendieck toposes, or the category of abelian sheaves on a small site.

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:

• any reflective subcategory of a total category

For example

• any category which is monadic over Set (see also (Kelly) Thm. 6.17)

• any category admitting a topological functor to Set

• The category Grp of groups as a category monadic over $Set$ 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 $C$ is total and $J$ is small, then $C^J$ is total, morally because it is a reflective subcategory of $Set^{C^{op} \times J}$; 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 $Set$ 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 $Grp \to Set$ would be representable. To see this is not the case, it suffices to produce a class of simple groups $G_\alpha$ of unbounded cardinality (for example, for any infinite set $X$, the alternating group $Alt(X)$, consisting of permutations of finite support that are even, is simple and of cardinality equal to that of $X$). For any group $G$, the hom-set $\hom(G_\alpha, G)$ consists of a single element (the trivial homomorphism) as soon as the cardinality of $G_\alpha$ exceeds that of $G$. Thus the class-indexed product $\prod_\alpha \hom(G_\alpha, G)$ is bounded in size, and defines a continuous functor $F = \prod_\alpha \hom(G_\alpha, -): Grp \to Set$. But it is clear this functor is not representable; e.g., for any group $G$, one can find $G_\alpha$ such that $F(G_\alpha)$ is much larger in size than $\hom(G, G_\alpha)$. This example is given in Wood 1982.

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

But cototal categories do occur:

• Set is cototal (as well as total).

• By dualizing Proposition , Ab is cototal (as well as total), because it is complete, well-powered, and has a cogenerator (e.g., $\mathbb{Q}/\mathbb{Z}$). Similarly, the category of modules $R Mod$ is cototal (and total) for any ring $R$. For that matter, any well-powered Grothendieck category, such as the category of abelian sheaves on a small site, is cototal.

• Similarly, the category $CH$ of compact Hausdorff spaces is cototal (as well as total, being monadic over $Set$), because like $Ab$ it is complete, well-powered, and has a cogenerator $I = [0, 1]$ (cf. Urysohn's lemma).

• If $C$ is cototal and $J$ is small, then $C^J$ is cototal.

• Any presheaf category of a small category is cototal (as well as total). Indeed, any Grothendieck topos is both cototal and total.

• Any category admitting a topological functor to Set is cototal (as well as total).

• Any totally distributive category is cototal (as well as total).

• Any coreflective subcategory of a cototal category is cototal, e.g., the category of compactly generated spaces is cototal.

Related pages

• totally distributive category

• lex total category

• large cocompleteness

References

• Syméon Bozapalides, Quelques Remarques sur les Cosmos Elementaires, Δελτίο της Ελληνικής Μαθηματικής Εταιρίας 15.15 (1974) 144-151 [pdf, pdf]

• Ross Street, Bob Walters, Yoneda structures on 2-categories, Journal of Algebra, 50 no. 2 (1978), p. 350-379, doi:10.1016/0021-8693(78)90160-6, (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

• Walter Tholen, Note on total categories, Bulletin of the Australian Mathematical Society cambridge journals

• 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)

• Ross Street, The family approach to total cocompleteness and toposes , Trans. A. M. S. 284 (1984) pp.355-369, (AMS)

• Richard J. Wood, Some remarks on total categories, J. Algebra 75 2 (1982) 538-545 [doi:10.1016/0021-8693(82)90055-2]

• Michael Albert Wendt. An introduction to totally cocomplete categories. Master thesis (1985). (url)

• Ross Street. The comprehensive construction of free colimits. Sydney Category Seminar Reports (Macquarie University). 1979. (pdf)

• Francisco Marmolejo, Robert Rosebrugh, and Richard Wood. Completely and totally distributive categories I. Journal of Pure and Applied Algebra 216.8-9 (2012): 1775-1790.

• Ross Street. Elementary cosmoi I. Category Seminar: Proceedings Sydney Category Theory Seminar 1972/1973. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.