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Stream: learning: questions

Topic: ✔ Connected components functor strong monoidal

Bruno Gavranović (Apr 02 2023 at 19:53):

Is the connected components functor $\pi_0 : \mathbf{Cat} \to \mathbf{Set}$ a strong monoidal functor (where we're using $\times$ as the monoidal structure for both $\mathbf{Cat}$ and $\mathbf{Set}$)?

I wager the answer is yes, since all the other adjoints $\mathsf{discr}, \mathsf{codiscr}$ and $\pi$ are strong monoidal, but I vaguely remember reading something about connected components being only lax. Is this true, or am I misremembering?

Todd Trimble (Apr 02 2023 at 20:38):

Yes, it's a reflexive coequalizer of product-preserving functors, where the parallel pair valued at a category $C$ is $\mathrm{dom}, \mathrm{cod} \colon C_1 \rightrightarrows C_0$. (Reflexive coequalizers are examples of sifted colimits.)

I think a lax monoidal functor between cartesian monoidal categories is automatically strong (notice colaxity comes for free for any functor, by taking advantage of cartesianness).

Bruno Gavranović (Apr 03 2023 at 09:09):

Ah, great, that makes sense.

Notification Bot (Apr 03 2023 at 09:10):

Bruno Gavranović has marked this topic as resolved.