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In this paper we see a Frobenius Monad in example 5.2. Suppose we take Hilb as the underlying category. Is this Frobenius Monad an internal category in [Hilb, Hilb]? If you can show that it is an internal category, please give some data about that category.

Heunen and Tull have a vaguely similar result in theorem 4.7.

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  • $\begingroup$ What is a "well recognizable" category? This seem not to be a usual technical term in category theory. It also does not occur in the preprint you are linking to. Would you please explain? $\endgroup$
    – Peter Heinig
    Sep 24, 2017 at 12:52
  • $\begingroup$ The question seems to presuppose that there's a natural way to regard an internal category in [Hilb, Hilb] as a Frobenius monad on Hilb. How does one do that? $\endgroup$
    – Tim Campion ♦
    Sep 24, 2017 at 13:30
  • $\begingroup$ Thank you for your comments. Perhaps I should remove the question about well recognizable. I just wanted some data about the internal category, if it existed. $\endgroup$
    – Ben Sprott
    Sep 24, 2017 at 15:08
  • $\begingroup$ I believe that the idea that a frobenius monad is also an internal category in [C,C] is an open question. Heunen did some work on something vaguely similar and I will add it to the question. $\endgroup$
    – Ben Sprott
    Sep 24, 2017 at 18:39
  • $\begingroup$ I would just comment that the structure of a monad on Hilb -- dagger Frobenius or otherwise -- relies on the composition monoidal product on [Hilb, Hilb]. Whereas the structure of an internal category in [Hilb, Hilb] has nothing to do with this monoidal product-- it just uses finite limits in [Hilb, Hilb]. So I don't see any connection between any kind of monad on Hilb and internal categories in [Hilb, Hilb]. The Heunen and Tull result is talking about monads in a bicategory of relations, which is very different from a monad in the bicategory Cat. $\endgroup$
    – Tim Campion ♦
    Sep 24, 2017 at 21:43

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Let me just expand on one aspect of my comment and recall the old chestnut that "most" bicategories whose objects are some kind of "structure" tend to fall into one of two classes:

  1. Bicategories whose 1-morpisms are "maps", e.g:
    • sets, functions, and identity 2-morphisms
    • rings, ring homomorphisms, and identity 2-morphisms
    • categories, functors, and natural transformations
    • ...

versus

  1. Bicategories whose 1-morphisms are "bimodules", e.g:
    • sets, spans, and maps of spans
    • sets, relations, and inclusion of relations
    • rings, bimodules, and bimodule homomorphisms
    • categories, profunctors, and natural transformations
    • ...

Very often, a bicategory of type 1 is included into a bicategory of type 2 to form a proarrow equipment, e.g.

  • (sets and functions) $\subset$ (sets and spans)
  • (sets and functions) $\subset$ (sets and relations)
  • (rings and homomorphisms) $\subset$ (rings and bimodules)
  • (categories and functors) $\subset$ (categories and profunctors)
  • ...

A monad in a type 1 bicategory often looks like, well, a monad. A monad in a type 2 bicategory often looks like some kind of category. Exercise: work out what a monad is in each of the above bicategories. In particular, a monad in $Span(C)$ is the same as an internal category in $C$.

The question is asking for a comparison between monads in a type 1 bicategory ($Cat$) and monads in a type 2 bicategory ($Span([Hilb,Hilb])$), which doesn't look very promising.

The linked comparison by Heunen and Tull compares monads in two type-2 bicategories (a category of relations vs. a category of spans), so passes the smell test.

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