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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”.
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.
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.
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.
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.
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).
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:
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 (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.
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
Walter Tholen, Note on total categories, Bulletin of the Australian Mathematical Society cambridge journals
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
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