total category




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.


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.


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.


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.


  • 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.


Any cocomplete and epi-cocomplete category with a generator is total. (And 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 of topological groups is total, as this 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. However, cototality is more rare. But cototal categories do occur:


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

Revised on June 11, 2017 09:41:14 by Todd Trimble (