Not-so-symmetric monoidal categories

Question

Is the notion of a commutative monoidal category, where every product is isomorphic to its opposite, but not necessarily functorially, known not to be useful? I have not been able to find any literature.

I have in mind endofunctors $\Sigma$ that are closed idempotents in the sense of Boyarchenko and Drinfeld: There is a natural transformation $\epsilon$ from the identity to $\Sigma$, such that $\epsilon \star \Sigma$ is an isomorphism and equal to $\Sigma \star \epsilon$.

Say that $( \Sigma , \epsilon )$ and $( \Sigma ' , \epsilon ' )$ commute up to isomorphism if there is an isomorphism $\gamma : \Sigma ' \Sigma \to \Sigma \Sigma '$ such that $\Sigma \star \epsilon ' = \gamma ( \epsilon ' \star \Sigma )$ and $\epsilon \star \Sigma ' = \gamma ( \Sigma ' \star \epsilon )$ .

In that case, $( \Sigma ' \Sigma , \epsilon ' \star \epsilon )$ is another closed idempotent.

It appears to me that such a condition of commuting up to isomorphism only involves the pair of endofunctors.

Answer 1

(This is really a comment, not an answer; but it's too long to fit in a comment.)

Carl, I'm unsure what you're asking. The first paragraph seems to be about monoidal categories such that $X \otimes Y \cong Y \otimes X$ for all objects $X$ and $Y$, but not naturally in $X$ and $Y$. The remaining paragraphs seem to be about something else entirely.

Perhaps the kind of example you had in mind in the first paragraph is the following. The category of finite totally ordered sets is monoidal: given two finite totally ordered sets $X$ and $Y$, the tensor product $X \oplus Y$ is the disjoint union of $X$ and $Y$ ordered by putting everything in $X$ before everything in $Y$. It has the property that $X \oplus Y \cong Y \oplus X$ for all $X$ and $Y$. On the other hand, there's no natural isomorphism between the functors $$ (X, Y) \mapsto X \oplus Y, \qquad (X, Y) \mapsto Y \oplus X, $$ so it's not a symmetric monoidal category.

But maybe that this is irrelevant to what you're thinking about. I'm afraid I just can't tell what you're after. Can you clarify?

Answer 2

I'm also not quite sure what the question is, but I'll answer anyway with the hope of saying something useful.

The notion of closed idempotent referred to above is the same as the notion of idempotent monad (a monad $(M, \epsilon: 1 \to M, \mu: M M \to M)$ whose multiplication $\mu$ is an isomorphism). Indeed, given closed idempotent data $(M, \epsilon: 1 \to M)$, define $\mu$ to be the inverse of $M\epsilon = \epsilon M$. Associativity of $\mu$ ($\mu \circ \mu M = \mu \circ M \mu$) is a consequence of $\mu M = M \mu$, which holds because both sides of the last equation are inverse to $M \epsilon M$.

The "intertwining constraint" $\gamma$ named in the question is a special case of a distributive law between monads. Generally speaking, a distributive law involves a natural transformation (not usually an isomorphism) $\theta: M M' \to M' M$ which is suitably compatible with the monad data on $M$ and $M'$, in such a way that the composite

$$M' M M' M \stackrel{M' \theta M}{\to} M' M' M M \stackrel{\mu' \mu}{\to} M' M$$

defines a monad multiplication on the composite $M' M$. (It seems clear that we need some compatibility conditions to be assured that this will work. The accepted distributivity conditions were worked out long ago by categorists, sometime in the mid-60's.)

(Readers might notice that a distributivity constraint $\theta$ looks a little like a braiding, and wonder if there's some connection there. In fact, Eugenia Cheng worked out such a connection in her paper on iterated distributive laws, which come up in the theory of higher categories, particular those which arise through a process of iterated enrichment. These iterated distributive laws involve Yang-Baxter equations which are formally like coherence conditions on a braiding.)

It sounds to me that Carl's question asks whether such intertwining constraints between functors, without some sort of extra compatibility or coherence conditions imposed, have been considered in the literature. (I think he must be abstracting away from the specific context of closed idempotents, since I can't imagine how to compose two such closed idempotents without some sort of compatibility with the $\epsilon$'s.) Anyway, I know of no such general theory of such things. I can't prove they are uninteresting, but the general trend in category theory is to pay close attention to coherence conditions, which pays off because they tend to have lots of mathematically interesting structure (as in the connection between monoidal categories and operations on 1-fold loop spaces, or interchange phenomena and iterated loop spaces, etc.).