*Thanks to @Eigil for catching a mistake in an earlier version of this (see comment below)*

Let \cal K be a monoidal closed 2-category with left Kan extensions and let T:\cal K \to \cal K be a lax monoidal endofunctor over it. Suppose V and A are T-algebras.

Often we want to define **T-convolution** of ‘functions’ A \to V, an operation that carries ‘T-terms’ of functions A\to V into new functions A \to V. In other words, a T-algebra structure on [A, V].

The paradigmatic example here is Day convolution, where \cal K= \bf Cat, V = \bf Set and T is the free monoidal category 2-monad. There, a ‘T-term of functions’ is a tuple of copresheaves A \to \bf Set over a monoidal category A. Day convolution gives you a new copresheaf on A from this data.

Another example is ‘Day convolaction’, which I previously described in Tambara modules are modules. In that case we have \cal K=\bf Cat and V=\bf Set again but T is the free \cal M-actegory 2-monad. Then Day convolaction endows [A,V] with an \cal M-action. (In fact it endows it with a whole [\cal M, V]-action, showing the following can be generalized further by replacing monads with graded monads, though, notably, Tambara theory only use the \cal M-action!).

The description of T-convolution is really simple. It’s made of three pieces:

- T is a lax monoidal functor, thus in particular lax closed, meaning there are coherent maps:

- V is a T-algebra, thus induces a map by post-composition:

- Finally, \cal K is closed so the T-algebra structure on A induces a map by left Kan extension:

Composing these maps gives you the desired T-algebra structure on [A,V].

An observation is: one could replace left Kan extension with any contravariant aggregation operation. This is especially useful when decategorifying the above, in which case one might replace the colimits involved in a Kan extension with e.g. sums in V. See pull-tensor-push.

Warning: the following is a bit speculative.

The above should work for {\cal K}={\bf Cat}/O, where O is a category of interfaces, and T is the 2-monad associated to a double operad \cal W of wiring operations with colours O. Now a T-algebra is a theory of systems indexed by \cal W. Let V be some other algebra, usually it’s something involving sets indexed by colours.

Then we can talk about \cal W-convolution of ‘quantities’ A \to V: given quantities (q_i:A(o_i) \to V(o_i))_{o_1, \ldots, o_n} and an operation w:o_1, \ldots, o_n \to o in \cal W, we can convolve the first along w to obtain w \ast (q_1, \ldots, q_n) : A(o)\to V(o).

Note that, crucially, we need \cal W to be double to be able to perform a Kan extension. In other words, we need to know how systems map into each other to know how to aggregate quantities on them.

I don’t know yet what I want to do with this operation but I suspect it might be useful to study compositionality of quantities defined over systems, chiefly behaviours.