nLab
total category

Definition

A locally small category $C$ is total if its Yoneda embedding $C\to [C^{\mathrm{op}},\mathrm{Set}]$ has a left adjoint. If $C^{\mathrm{op}}$ is total, $C$ is called cototal.

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

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 $F:D\to C$ is a functor such that ${\lim }_d{\mathrm{Hom}}_C(X,Fd)$ is a small set for all $X\in C$, then $F$ has a limit.

Examples

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:

• locally presentable categories, hence in particular Grothendieck toposes.

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

• any category which is monadic over Set

• any category admitting a topological functor to Set

References

• 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)