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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.
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 $F\colon D\to C$ 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.
Any cocomplete and epi-cocomplete category with a generator is total. (And 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 admitting a topological functor to Set
The category of topological groups is total, as this 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. However, cototality is more rare. But cototal categories do occur:
Set is cototal (as well as total).
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.
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