monoidal structures on functor categories

Question

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.

Answer 1

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.