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1. • CommentRowNumber1.
   • CommentAuthorBen_Sprott
   • CommentTimeNov 5th 2017
   • (edited Nov 5th 2017)
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   Author: Ben_Sprott
   Format: MarkdownItexHi, I [asked this question](https://mathoverflow.net/q/283633/10007) on math overfow. It was about Bimonads on Set that are exact. Todd Trimble gave a comment about the following endofunctors: $M \circ -$ and $- \circ M$. I am now very curious about these. What are these Bimonads? What are the concrete examples for their natural transformations? What is the source category for the adjunction that gives rise to these Bimonads on Set?

   Hi,

   I asked this question on math overfow. It was about Bimonads on Set that are exact. Todd Trimble gave a comment about the following endofunctors: $M \circ -$ and $- \circ M$.

   I am now very curious about these. What are these Bimonads? What are the concrete examples for their natural transformations?

   What is the source category for the adjunction that gives rise to these Bimonads on Set?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 5th 2017
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   Author: Mike Shulman
   Format: MarkdownItexAs Todd said, he wasn't answering your question, he was explaining what he thought your question meant. $M$ is the bimonad you're wondering about the existence of, on some category $C$, and the property that $(M \circ -) : [C,C] \to [C,C]$ and $(- \circ M) : [C,C] \to [C,C]$ preserve equalizers (in the endofunctor category, which are calculated pointwise if $C$ has equalizers) is (he thought) the property that you were asking whether a bimonad $M$ could have.

   As Todd said, he wasn’t answering your question, he was explaining what he thought your question meant. $M$ is the bimonad you’re wondering about the existence of, on some category $C$, and the property that $(M \circ -) : [C,C] \to [C,C]$ and $(- \circ M) : [C,C] \to [C,C]$ preserve equalizers (in the endofunctor category, which are calculated pointwise if $C$ has equalizers) is (he thought) the property that you were asking whether a bimonad $M$ could have.

3. • CommentRowNumber3.
   • CommentAuthorBen_Sprott
   • CommentTimeNov 6th 2017
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   Author: Ben_Sprott
   Format: MarkdownItexThanks Mike!

   Thanks Mike!