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Stream: theory: category theory

Topic: funny tensor product of categories

Jem (May 27 2020 at 19:32):

The funny tensor product $C \square D$ is a monoidal product on $Cat_{str}$ defined by the pushout of $C_0 \times D \leftarrow C_0 \times D_0 \to C \times D_0$, where $C_0$ and $D_0$ are the discrete categories on the objects. This forms a symmetric monoidal closed structure on $Cat_{str}$, with the internal hom defined to be the category of functors and unnatural transformations (i.e. transformations without the naturality condition).
Unfortunately, the funny tensor product isn't invariant under equivalence of categories. The unit is $\mathbf 1$ so $\mathbf 1 \square \mathbf 1 \cong \mathbf 1$, but denoting the indiscrete category on two objects $\mathbf I$, we get that $\mathbf I \square \mathbf I \cong B\mathbb Z$ is a (four object) category equivalent to the delooping of the integers.
Is there a variant of this tensor product which is invariant under equivalence? We could replace $C_0$ with $core(C)$ above, but the nlab states that this won't be closed in the 1-category $Cat_{str}$ (see Algebraic categories with few monoidal biclosed structures or none by F. Foltz, G.M. Kelly, and C. Lair.). What happens in the 2-categorical case (or the 1-category of non-strict categories, for that matter)?

John Baez (May 27 2020 at 19:50):

The "funny tensor product" of 1-categories is the goofy younger brother of the Gray tensor product of 2-categories.

Mike Shulman (May 28 2020 at 17:24):

Jem said:

the nlab states

Where? I don't see it on the page funny tensor product.

Jem (May 28 2020 at 17:34):

It's on that page, but not stated explicitly:

This tensor product □ is called the funny tensor product. This constitutes one of the precisely two symmetric monoidal closed structures on Cat; the other of course is the cartesian closed category structure on Cat.

Mike Shulman (May 29 2020 at 02:06):

Ah, I see.

Mike Shulman (May 29 2020 at 02:07):

Non-strict categories don't form a 1-category. I don't think there's much chance of the core-variant working in the 2-category Cat, but I would give it reasonable odds of working in the 2-category $\mathrm{Cat}_g$ of categories, functors, and natural isomorphisms. I don't think I've ever seen anyone try it though.

Jem (May 31 2020 at 12:49):

Ah, thanks for the reply.

I am confused about something though - don't the non-strict categories form a 1-category when we take morphisms to be functors, identified up to the existence of a natural isomorphism between them?

Mike Shulman (Jun 03 2020 at 08:54):

Well, yes; that's the homotopy 1-category of $\mathrm{Cat}_g$. But it would be misleading to call that "the 1-category of categories".