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Ben Sprott (Nov 19 2020 at 04:14):Here we see Spivak defining Categories as comonoids in the monoidal category , section 2.5. Comonoids in an endofunctor category are comonads. I would guess that if you add a monad structure to your comonoids, you still have a category, but you have extra structure. It would be interesting to know what extra stuff you are endowing the category with if you add monad structure.
Jade Master (Jan 28 2021 at 01:35):@Ben Sprott coud you clarify what your question is? Are you adding a monad structure to comonoids in Poly? If so I'm not sure what you mean by monad structure...maybe you mean a monoid structure?
Ben Sprott (Jan 28 2021 at 02:12):@Jade Master said
@Ben Sprott coud you clarify what your question is? Are you adding a monad structure to comonoids in Poly? If so I'm not sure what you mean by monad structure...maybe you mean a monoid structure?
I apologise in advance, but this might not make sense. I believe that Spivak is highlighting comonads on Set as being internal Categories in Poly. This is so because an internal category in a monoidal category is a comonoid in that category. Furthermore, since the monoidal category in question here is an endofunctor category, these comonoids are comonads on Set. What I am proposing is that we highlight, instead, objects in Poly that are both comonoids and monoids, so that they are actually both monads and comonads. I am guessing that these objects are, again, categories, but they have added structure. I wondering what that extra structure is. So the equation is twofold:1) am I making sense 2) what is the extra structure as mentioned above.
Ben Sprott (Jan 28 2021 at 02:20):I have been trying to get help with this for six years
Jade Master (Jan 28 2021 at 02:24):A comonad or monad is something that only makes sense in 2-category. For a 2-category C, a monad in C is a monoid in a hom-category C(x,x).
Jade Master (Jan 28 2021 at 02:26):Sorry you've been looking for an answer for so long :0
Jade Master (Jan 28 2021 at 02:29):An internal category in a monoidal category C is not the same as a comonoid in C. A comonoid in C is an object x equipped with a coassociative counital operation
Jade Master (Jan 28 2021 at 02:35):After reading the link on your overflow question I see that it's talking about comonads in C, when it is regarded as a one object bicategory. So for any monoidal category C, there's a bicategory of comonads in C, CoMonad(C). Then the nlab says that internal categories are monads in CoMonad(C). Mind that I have no idea how this ends up being anything like a category, but this is what the nlab article says.
John Baez (Jan 28 2021 at 02:42):Take C = Set. Is the nLab claiming that a category in Set is also a monad in CoMonad(Set)?
It reminds me of how a category in Set is a monad in Span(Set).
Ben Sprott (Jan 28 2021 at 02:44):Just to be clear, let's paste the link that I was using. That's the nlab article on internal categories in a monoidal category.
Nathanael Arkor (Jan 28 2021 at 02:48):I think the confusion arises because Spivak is not considering comonads on Set as internal categories in Poly. The comonoids in Poly are not internal categories in the sense of that nLab page. They're internal categories in Set.
Nathanael Arkor (Jan 28 2021 at 02:49):However, one could still consider the comonoids in Poly (i.e. categories) that are also monoids (perhaps compatibly with the comonoid structure somehow).
Nathanael Arkor (Jan 28 2021 at 02:51):This is definitely a reasonable question, and I imagine the answer isn't obvious: one would need to sit down and work it out.
Nathanael Arkor (Jan 28 2021 at 02:53):Asking for internal categories in categories of functors is a different question, but one for which you probably don't need the monoidal structure. If you consider a category of endofunctors with finite limits, you can define an internal category in the usual way. (But I see you've already received an answer to this question.)