This post originally appeared on LocalCharts.
Thanks to Eigil Fjeldgren Rischel for catching a mistake in an earlier version of this post (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: \[T[A,V] \to [TA, TV]\]
- \(V\) is a \(T\)-algebra, thus induces a map by post-composition: \[[TA,TV] \to [TA, V]\]
- Finally, \(\cal K\) is closed so the \(T\)-algebra structure on \(A\) induces a map by left Kan extension: \[[TA,V] \to [A,V]\]
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.