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Stream: learning: questions

Topic: morphisms of cofunctors

Mike Stay (Apr 17 2022 at 21:19):

Is there a usual notion of morphism between cofunctors? nLab's got nothing on "conatural transformation". This tweet suggests thinking at cofunctors as spans, so I guess it's conceivable one could consider maps of spans or spans of spans.

A cofunctor is a span of functors, whose left leg is a bijective-on-objects functor, and whose right leg is a discrete opfibration. For a fixed category B, there is a category Cof(B) of cofunctors over a base, and DOpf(B) is a coreflective subcategory of it. (6 / 11)

- Bryce Clarke (@8ryceClarke)

John Baez (Apr 17 2022 at 23:08):

Maybe @Bryce Clarke can say.

Tim Hosgood (Apr 18 2022 at 19:54):

There's a definition on slide 17 of Bryce's talk from CT2019 (https://conferences.inf.ed.ac.uk/ct2019/slides/63.pdf)
Screenshot-2022-04-18-at-21.53.40.png

Mike Stay (Apr 18 2022 at 20:41):

Thanks! Maps of spans it is, then. Unless I'm missing something, the cofunctors are special spans in Cat, so the resulting structure Cof should be a sub-tricategory of the monoidal tricategory Span(Cat) described by @Alex Hoffnung here. The diagram you posted says transformations between cofunctors are maps of spans where the opfibration part commutes on the nose. Because it commutes strictly, that allows only identity modifications between transformations.

@Bryce Clarke, is there some reason you didn't consider maps of spans with 2-cells on both sides and the resulting modifications?

Bryce Clarke (Apr 19 2022 at 06:55):

Mike Stay said:

Thanks! Maps of spans it is, then. Unless I'm missing something, the cofunctors are special spans in Cat, so the resulting structure Cof should be a sub-tricategory of the monoidal tricategory Span(Cat) described by Alex Hoffnung here. The diagram you posted says transformations between cofunctors are maps of spans where the opfibration part commutes on the nose. Because it commutes strictly, that allows only identity modifications between transformations.

Bryce Clarke, is there some reason you didn't consider maps of spans with 2-cells on both sides and the resulting modifications?

Hi Mike, there are a few reasons why these specific maps were considered between cofunctors (viewed as spans). One reason, is that this is how natural cotransformations were defined in Aguiar's thesis p39. The second and more important reason is that these are the kind of 2-cells which arise naturally from the formal theory of monads II. For a 2-category K, there is a 2-category EM(K) and when we restrict to morphisms of monads whose 1-cell component is a left adjoint, you get the 2-category of categories, cofunctors, and conatural transformations.