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Stream: learning: questions

Topic: conditions for points functor to have a left SMC adjoint?

Mike Stay (Feb 22 2021 at 18:17):

Given a symmetric monoidal closed category $(V, \otimes, I, \multimap),$ under what conditions does the "points" forgetful functor $V(I, -):V \to {\rm Set}$ induce a symmetric monoidal closed adjunction?

Martti Karvonen (Feb 22 2021 at 18:35):

If V has coproducts, then the functor has a left adjoint (This is e.g in Borceux Vol 2 prop 6.4.6). I don't know if this left adjoint is a SMC-functor without further conditions though.

Mike Shulman (Feb 22 2021 at 19:01):

The left adjoint takes copowers by the unit object, $F(X) = X\cdot I$. This is strong symmetric monoidal whenever $V$ is closed (with coproducts), since then the tensor product preserves colimits (including copowers) in each variable.

Mike Stay (Feb 25 2021 at 18:14):

Thanks!