Added a reference to a paper of Bozapalides on total objects in a 2-category.

]]>fixed ref

- {#StreetWalters}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)

I rewrote the Idea section, including a quote by Street, to help give an inkling of just what sort of category one can expect will be total.

]]>Mention “absolutely cocomplete” terminology, though I think this could be confusing, since “absolutely cocomplete” could refer to categories having absolute colimits.

]]>I have tried to fix the grammar in the first sentence, which globally went as: “A total category is … but admitting most types of categories…”

(It’s instead the *notion* of total category which can admit certain examples.)

Also changed “most” to “many”.

]]>Add another characterisation of totality.

]]>Add reference to *An introduction to totally cocomplete categories*.

Added link to Kelly reference, and added some other links.

]]>I added a reference for proof that categories monadic over Set are total; maybe someone can make the reference into a link to the paper by Kelly at the bottom of the page.

]]>The fact that total implies complete is a categorification of the fact that for partial orders cocomplete implies complete.

]]>corrected year of Ross Street’s publication and inserted a link to the article at the AMS journal website

]]>I added some more examples to total category. (One is that Ab is cototal as well as total.)

]]>Yes, good observation.

It’s known for example that $Grp$ is not cototal, and neither is say the category of commutative rings $CRing$. An easy way to see this is to produce continuous functors $C \to Set$ that are not representable, e.g., for $C = Grp$, the classical example is the class-indexed product of representables $hom(G,-)$ where $G$ ranges over all simple groups. (For any group $H$, $Hom(G, H)$ will be trivial once the simple group $G$ has cardinality greater than $H$, so the product of $Hom(G, H)$ over all simple $G$ will still be a set.) A similar example can be cooked up for commutative rings; see e.g. this MO answer. I guess algebraic categories with a plentiful supply of simple objects would be amenable to similar constructions.

]]>The article currently says something to the effect of “cototal categories are more rare than total categories”. But it occurs to me that $Top$ is cototal by Day’s criterion (it’s complete, mono-complete, and has a cogenerator given by the indiscrete space on two elements). In fact, since being a topological functor is self-dual, and since $Set$ is cototal, any category which is topological over $Set$ is cototal – I’ll add this to the article as a class of examples. I don’t know of a reason to expect categories of a more “algebraic” nature to be cototal, but at least this suggests that many categories of “spaces” might be cototal.

]]>at *total category* I have added after the definition and after the first remark these two further remarks:

+– {: .num_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.

=–

+– {: .num_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*.

=–

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