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• CommentRowNumber1.
• CommentAuthorBryceClarke
• CommentTimeOct 1st 2020

Created a stub for cofunctor? with some references.

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeOct 2nd 2020

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 2nd 2020

I suggest to include author names in double square brackets only if the author page exists or if you create it right away. Because otherwise the links will just remain broken forever and look bad.

Here Mackenzie’s page did exist, and I added the redirect to it which makes the requested link work here.

For the last reference I removed the double square brackets on the author names. But feel invited to add them back in – together with the respective author pages.

• CommentRowNumber4.
• CommentAuthorBryceClarke
• CommentTimeJul 3rd 2021

Added in definition of cofunctor, attempting to mimic to external definition of functor. The use of $\varphi_{0}$ and $\varphi_{1}$ feels a bit clunky, so there may be a more appropriate choice of notation.

• CommentRowNumber5.
• CommentAuthorBryceClarke
• CommentTimeJul 3rd 2021

Made minor changes to definition, added Tikz diagram as a visual aid. Added definition of composition of cofunctors, and noted that small categories and cofunctors form a category. Added several basic examples. I also added a couple of references to my own papers on cofunctors.

I’d like to make a note somewhere there there are two different conventions in the literature for the direction of a cofunctor. Specifically, this page uses the convention that cofunctors are defined in the same direction as the object assignment, whereas other references (in particular Aguiar 1997, Definition 4.2.1 where cofunctors were first defined) define cofunctors in the direction of the morphism assignment. Where would the most appropriate place to put this note?

• CommentRowNumber6.
• CommentAuthorBryceClarke
• CommentTimeJul 4th 2021

Created a Properties section and added a proposition stating the orthogonal factorisation system on Cof. Full proof will be filled in later. Added further examples of cofunctors. Added further references (papers by Garner) and some brief comments about each.

I’m also wondering: there must be a link to the notion of comorphism, but it is not obvious to me. Does anyone have an idea? There are a variety of notions of comorphism in the Higgins-Mackenzie paper, so perhaps that provides is related.

• CommentRowNumber7.
• CommentAuthorBryceClarke
• CommentTimeAug 9th 2021
• (edited Aug 9th 2021)

Added link to Spivak-Niu draft textbook chapter on cofunctors, and the proposition that $\mathbf{Cof}$ is the category of comonoids in $\mathbf{Poly}(1, 1)$.