## Montag, 29. Juli 2013

### Categorial constructions (Part 1)

Reading up on a paper, I discovered that I will need Day's convolution to construct tensors in a bicategory of monoidal supported profunctors. The hom-categories of these to $Set$ then capture supported pre-categories as a partial monoid.

So slowly working toward it, I flesh out some of my notes here:

Usually, one encounters convolution in electrical engineering and in image processing; the convolution of two functions $f$ and $g$, denoted by the convolution operator $*$ as $(f * g)$ is defined as

$(f*g)(t) = \int_{-\infty}^{\infty} f(\tau) f(t - \tau) d \tau$

where $f, g: M \rightarrow \mathbb{C}$ are maps from a group to complex numbers. It can be thought of as a moving window, giving by one of the function, that inspects the values of the other function. This definition can be extended to convolutions on functors $F$ on a monoidal category $M$ to the category of sets $Set$, $F: M \rightarrow Set$.

In the standard setting, convolution yields a commutative algebra without identity on the linear space of (suitably) measureable functions. For the extension, the functor category $Set^M$ of functors $F: M \rightarrow Set$ yields a monoidal category; Day's convolution is its tensor product.

A simple example to see how it works is given by graded sets in a categorical setting: given the discrete category $\mathbb{N}$ with natural numbers as objects and only identify mappings as morphisms, where addition plays the role of the monoidal product, the graded sets then can be represented by functors $F: \mathbb{N} \rightarrow Set$. The convolution in the category then is

$(F*G)(n) = \sum_{i+j=n} F(i) \times G(j) = \sum_{i,j} F(i) \times G(j) \times hom_{\mathbb{N}(n, i \otimes j)$

Where $\otimes$ is the addition of natural numbers. Moving on to a more general setting, one wants to replace $\mathbb{N}$ by a general monoidal category $C$. The formula for the convolution, given two presheaves $F, G: C^{op} \rightarrow Set$ is defined as

$(F*G)(e) = \int^{c,d \in Obj(C)} F(c) \times G(d) \times hom_C(e, c \otimes d)$

In that expression, the convolution product is given by a coend $\int$ of the functor under the integral symbol. The coend of an functor $F$ in $Set$ is an object $e$ with a universal dinatural transformation $\zeta: F \stackrel{..}{\rightarrow} e$. In this case, the notion of dinatural transformation relaxes the requirements of a natural transformation: usually, for a natural transformation $\alpha: F \rightarrow G$ between functors F and G, $\alpha$ depends on some variable $x$ co-/contravariantly. A dinatural, or in this case the slightly more restrictive notion of an extranatural transformation requires either F or G to depend on some variable both co- and contravariantly. It is constituted by a collection of morphisms $\alpha_c: F(c,c) \rightarrow G(c,c)$ such that for every morphism $f: c \rightarrow c'$ in $C$ the hexagon identity holds:

$G(c,f)\alpha_cF(f,c) = G(f,c')\alpha_{c'} F(c',f):F(c',c) \rightarrow G(c,c')$

In this setting, the coend object is an object of $Set$. The Day's convolution has some nice properties regarding the preservation of colimits, as well as some interesting relations to the Yoneda embedding. The next posts will contain more on the properties on the convolution and possibly the Yoneda lemma, but also mainly focus on recovering supported precategories in a more abstract setting.