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I am wondering if it is possible to have a bimonad on $\mathsf{Set}$ that preserves equalizers on both sides? What about a bimonad that is exact? Can you give an example?

Let me try to explain what I mean by preserving equalizers on both sides. One of the reasons I am interested in Bimonads, is that I think they may be internal categories in an endofunctor category on Set. Have a look at this post.

I want to interpret the monoidal product $\otimes$ as functor composition. In the link, they require that the monoidal product preserve equalizers on both sides. This means that we want the functor composition to preserve equalizers. I am not actually sure what it would mean for the functor composition to preserve equalizers.

A bimonad is a monad that is a comonad, and these interact according to a mixed distributive law, as described by Mesablishvili and Wisbauer here.

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  • $\begingroup$ What do you mean by on both sides? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 17, 2017 at 4:21
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    $\begingroup$ @მამუკაჯიბლაძე Based on past questions, I think Ben means that both $M \circ -$ and $- \circ M$ preserve equalizers (but of course the latter is automatic since equalizers are computed pointwise). Also, I'll bet that reflexive equalizers would be enough for his purposes. $\endgroup$
    – Todd Trimble
    Commented Oct 17, 2017 at 10:43
  • $\begingroup$ @ToddTrimble what is the Monad you are describing? I have a question here about something similar. Is it the same? $\endgroup$
    – Ben Sprott
    Commented Nov 5, 2017 at 16:18
  • $\begingroup$ I wasn't describing any monad in particular; I was just interpreting for მამუკაჯიბლაძე what I thought you meant by "preserves equalizers on both sides". $\endgroup$
    – Todd Trimble
    Commented Nov 5, 2017 at 17:30
  • $\begingroup$ Let me write what I mean in the question. $\endgroup$
    – Ben Sprott
    Commented Nov 5, 2017 at 19:14

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