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Stream: theory: applied category theory

Topic: Examples of Strict monoidal functors

Kobe Wullaert (Jun 09 2022 at 12:49):

Hi everyone. Do you know of any (interesting) examples of forgetful functors between monoidal categories which are strict monoidal and in particular examples of sections of such functors?

Simon Burton (Jun 09 2022 at 20:16):

How about the monoidal functor that takes the underlying vector space of a group representation?

John Baez (Jun 10 2022 at 02:16):

That's a nice one. In general forgetful functors that only forget structure or properties can be made strict monoidal, since the monoidal structure on the 'stuff' is unchanged. For example

$\mathsf{Grp} \to \mathsf{Set}$

just forgets the group operations, which are structure, while leaving the underlying stuff - the set - unchanged. So, if we make $\mathsf{Set}$ monoidal by choosing a specific cartesian product for each pair of sets, and use this same choice to make $\mathsf{AbGp}$ monoidal, the forgetful functor painlessly works out to be strict monoidal.

John Baez (Jun 10 2022 at 03:20):

The functor

$\mathsf{AbGp} \to \mathsf{Gp}$

which just forgets the property of being abelian, can also be arranged to be strict monoidal in a similar way.

Mike Shulman (Jun 11 2022 at 00:15):

There are lots of examples like that, some of which also have sections. For instance, we can do the cartesian product trict for the forgetful functor

$$\rm Top \to Set$$

and this has two sections that equip a set with the discrete or the indiscrete topology, respectively.