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Stream: learning: questions

Topic: Thing that distributes over monoid morphisms

Asad Saeeduddin (Sep 09 2020 at 05:15):

Bit of a shot in the dark, but does anyone know of a name for a construction involving:

• a monoidal category $C, \otimes, I$
• monoids $A, B : Mon(C)$
• morphisms $f : U(A) \to U(A)$ and $g : U(B) \to U(B)$

such that for any morphism $h : Mon(C)(A, B)$:

$$g \circ U(h) = U(h) \circ f$$

(or weaker in the case of a bicategory, e.g. up to 2-isomorphism or just 2-morphism)

Asad Saeeduddin (Sep 09 2020 at 05:24):

in the case of it being weaker there's probably some desirable coherence conditions, but maybe someone here can just spot what this is right off the bat and save me the trouble of having to grope about for what they should be

Morgan Rogers (he/him) (Sep 09 2020 at 09:00):

If you extended $f,g$ to a morphism on every monoid, they would assemble into a natural transformation $U \Rightarrow U$ where $U:\mathbf{Mon}(\mathcal{C}) \to \mathcal{C}$. So you could take the full subcategory of $\mathbf{Mon}(\mathcal{C})$ on $A$ and $B$ and restrict $U$ to it, although that seems less natural than trying to extend this to a natural transformation on all of $\mathbf{Mon}(\mathcal{C})$; it depends how flexible these things need to be.

Asad Saeeduddin (Sep 09 2020 at 18:19):

@[Mod] Morgan Rogers Sorry, I don't quite follow what you mean by extending $f, g$ to a morphism on every monoid.

Morgan Rogers (he/him) (Sep 10 2020 at 10:05):

I mean that if I write $\alpha_A$ for $f$ and $\alpha_B$ for $g$, they look just like two components of a natural transformation. The equation you give is the naturality condition (but just for these two components of the transformation)