First of all, I'm not an expert in category theory, but I have the following question:
Are there any other known monoidal structures on functor categories besides the pointwise tensor product and the Day convolution product?
Thank you.
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$\begingroup$ This kind of thing often depends on the structure of the codomain category. $\endgroup$– RandallOct 25, 2017 at 15:54
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$\begingroup$ If you have a monoid object in a functor category, you can make the slice category (over the monoid object) into a monoidal category. If you can make a monoid out of the terminal functor in some category, then obviously the slice over the terminal object is the same as the category itself... At least for sheaf toposes this should be different from the cartesian monoidal product anyways $\endgroup$– neptunOct 25, 2017 at 16:10
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I'll interpret the question as follows: is there an example of a monoidal structure on some functor category $[C, D]$ which does not arise as either the pointwise monoidal structure wrt some monoidal structure on $D$, or as Day convolution wrt some monoidal structures on $C$ and $D$?
(That's one version of the question. Another version is whether there are any such constructions which are furthermore natural in $C$ and $D$, possibly after equipping them with extra structure.)
Here's one very nice example. Let $C$ be the (opposite of, but it doesn't really matter) groupoid of finite sets and bijections, and let $D$ be the category of sets. Then $[C, D]$ is the category of combinatorial species. This category supports many monoidal structures which arise from the operations you've described already, for example Day convolution wrt disjoint union or cartesian product on $C$, and pointwise product and coproduct.
It also supports a monoidal structure which does not arise in this way called the substitution or composition product. One way to define it is as follows: a species $F$ acts on the objects of any symmetric monoidal cocomplete category $(C, \otimes)$ (this includes the condition that the symmetric monoidal structure distributes over colimits) as a "power series"
$$C \ni X \mapsto \sum_{n \ge 0} F_n \otimes_{S_n} X^{\otimes n}$$
and the substitution product comes from composing these operations; this is a categorification of composition of formal power series. Monoids wrt the composition product are the same thing as (symmetric) operads.
answered Oct 25, 2017 at 17:33