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Stream: theory: category theory

Topic: explicit description of the lax monoidal functor classifier?

Martti Karvonen (Jan 09 2022 at 17:56):

The famous BPK-paper guarantees that the inclusion of (strict) monoidal categories and strict monoidal functors into (strict) monoidal categories and lax monoidal functors has a left adjoint $L$. Hence the category of lax monoidal functors $C\to D$ is isomorphic to that of strict monoidal functors $L(C)\to D$. Is there a known explicit construction of $L$ in this case? For instance, I know that when $1$ is the terminal monoidal category, then $L(1)$ is the free monoidal category on a monoid, but I'd love to be able to compute $L(-)$ more generally.

Amar Hadzihasanovic (Jan 09 2022 at 20:09):

See this reply by Rune Haugseng for the more general case of lax functors of 2-categories (just specialise to the “single object” case).

Martti Karvonen (Jan 09 2022 at 20:32):

Thanks, that does it!