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BTW congrats on your paper!
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.
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?
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.
Mention “retrofunctor” terminology. From people I’ve spoken to, there is general consensus that this is a more appropriate, and less misleading, name than “cofunctor”. However, it probably isn’t appropriate to rename this page until there are more references using this terminology.
Rename page to “retrofunctor” rather than “cofunctor”, and added a note on terminology. Although this usage is still nascent in the literature, it is increasing in popularity (see for instance recent talk slides by Bryce Clarke [1] [2]), and avoids the misleading connotations of the “co-” prefix in “cofunctor”. Furthermore, it matches the terminology of monad retromorphism.
Mention cocategories in relation to the “cofunctor” terminology.
I hope it’s ok to remove a few instances of ” where $a' = cod(\varphi_{1}(a, u))$”. I’ve read this page many times and always get confused by it. It seems like saying “$f:X\to Y$ where $Y = cod(f)$” - it doesn’t add any new information and just trips me up wondering why it’s mentioned.
Nathaniel Virgo
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