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Stream: theory: category theory

Topic: Nerve of a strong monoidal functor is lax monoidal

Max New (Sep 20 2024 at 16:59):

Let $C$ be a monoidal category and $F : C \to D$ a strong monoidal functor. I need for a gluing construction I'm using that the nerve functor $D(F(=),-) : D \to Psh(C)$ is lax monoidal but I'll admit this looks a bit tedious to verify the equations. It seems like a common enough situation does anyone have a reference that proves this?

Max New (Sep 20 2024 at 17:16):

Since Yoneda is strong monoidal this reduces to showing that precomposition with a strong monoidal functor is lax monoidal on the Day convolution

Nathanael Arkor (Sep 20 2024 at 17:30):

I think this follows from Proposition 54 of Walker's Distributive laws via admissibility, together with Remark 56, taking $T$ to be the free strict monoidal category 2-monad, and $P$ to be the free cocompletion.