Recall that a locally small category is total if its Yoneda embedding has a left adjoint .
A total category is totally distributive if has a further left adjoint .
If is totally distributive, then since is fully faithful, then, by the properties of adjoint triples, so is . Thus, is a coreflective subcategory of , which is cototal (or more precisely, pro-cototal?, since it is not locally small) β hence is also cototal.
Moreover this means that the induced adjoint pair of (co-)monads
is an adjoint modality.
If is a small category, then its presheaf category is totally distributive.
If is totally distributive and is a full subcategory that is both reflective and coreflective, then is totally distributive.
By the adjoint functor theorem for total categories, a total category is totally distributive if and only if the functor preserves all limits. Moreover, for any total category, it is possible to define the functor which βwants to beβ the left adjoint of (and is if is total). The elements of are called waves from to (just because they are usually denoted by wavy arrows), and as in the case of continuous categories they form an idempotent comonad on in the bicategory of profunctors.
Totally distributive categories are βalmostβ an example of continuous algebras for a lax-idempotent 2-monad.
Robert Rosebrugh, Richard J. Wood, An adjoint characterization of the category of sets. PAMS 122 2 (1994) 409-413 [jstor:2161031]
Rory Lucyshyn-Wright, Totally distributive toposes. spnet.
Francisco Marmolejo, Bob Rosebrugh, and Richard Wood, Completely and totally distributive categories I, JPAA 216 no. 8-9 (2012). PDF
Last revised on April 19, 2023 at 09:12:29. See the history of this page for a list of all contributions to it.