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Stream: learning: questions

Topic: pseudomonoid objects

Beppe Metere (Oct 14 2022 at 06:35):

Let $\mathcal{C}$ be a category with finite products. One can define a monoid object internal to $\mathcal{C}$ as an object $M$ of $\mathcal{C}$ endowed with identity $1\to M$ and multiplication $M\times M\to M$ satisfying a bunch of diagrams. However, one can also define a monoid object in a representable fashion: a functor $F\colon \mathcal{C}^{op}\to \mathbf{Mon}$ such that the composite with the forgetful functor $\mathbf{Mon}\to \mathbf{Set}$ gives the representable functor $h_M$.

It seems to me that the "same" can be done in dimension 2. Am I right? I mean, a pseudomonoid object in a 2-category is usually defined as an object with structure satisfying some conditions, which are expressed as diagrams filled with coherent isomorphisms. Should one also define them as pseundofunctors to $\mathbf{MonCat}$ that composed with the forgetful 2-functor $\mathbf{MonCat}\to \mathbf{Cat}$ give 2-representables?

Did anyone ever take this point of view?

Beppe Metere (Oct 18 2022 at 07:35):

hmmm... so this lack of answers means this point of view has never been studied before, maybe just because it is not interesting?

Mike Shulman (Oct 18 2022 at 16:31):

You're certainly right that it can be formulated this way. I don't know of any references, but rather than it not being interesting, my guess is it's just that no one has found a use for it.