admin – Page 2 – Profunctors

January 5, 2018October 31, 2022

Plenty faces of a profunctor

What is a profunctor?

Recall that a category is a directed graph equipped with an associative and unital composition of its edges. In the categorical context, the nodes of the graph are also called objects and edges are called arrows or morphisms. The unitality requirement says that each object $a$ has a unit arrow $a\to a$ , called identity, which doesn’t affect the result of the compositions.

A functor is a composition and identity preserving graph morphism from a category to a category.

A profunctor, on the other hand, connects up two categories by (potentially) adding more morphisms in between them ‘from the outer world’. In other perspective, we can describe this setting as one of the two categories acting from the left and the other one acting from the right on the newly added morphisms. This can also be grasped as simply being a two variable functor to the category of sets, contravariant in the first argument.
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January 1, 2018November 9, 2022

Functor as a fibration

In topology, any continuous function $f:A\to B$ can be considered as a fibration over space $B$ , with total space $A$ , where the fibres $f^{-1}(b)$ are glued together according to the topology of $A$ .

Likewise, any functor $F:\ct A\to \ct B$ can be regarded as a kind of fibration over $\ct B$ . Each object $b\in Ob\ct B$ determines a fibre category $F^{-1}(b)$ with arrows which are mapped to $1_b$ .

In this generality, the fibres connect up by means of profunctors between them and then $F$ and the original composition of $\ct A$ are encapsulated in the arising mapping $\ct B\to \ct Prof$ .

November 29, 2017November 9, 2022

On the two definitions of adjunctions

(Note that we write compositions from left to right and accordingly apply most maps on the right.)

Def.1
A functor $F:\ct A\to\ct B$ is called a left adjoint to a functor $U:\ct B\to\ct A$ if there is a bijection between the homsets:

$\ct B(A^F\ig B) \simeq \ct A(A\ig B^U)$

natural in both $A$ and $B$ . In this case $U$ is called a right adjoint to $F$ , and we write $F\adj U$ .

Def.2
A functor $F:\ct A\to\ct B$ is called a left adjoint to a functor $U:\ct B\to\ct A$ if there are natural transformations $\eta:1_{\ct A}\tto FU$ and $\eps:UF\tto 1_{\ct B}$ satisfying the zig-zag identities:

$\matrix{\eta F\cdot F\eps = 1_F &\sep U\eta\cdot \eps U=1_U}$

So, why are these two definitions equivalent?
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November 28, 2017

Morita equivalence by categorical bridges

Let $\mathcal Iso$ denote the category of 2 objects (name them $0,1$ ) and 2 nonidentity arrows, $f:0\to 1,\ g:1\to 0$ which are inverses of each other.

We define a bridge as a category over $\mathcal Iso$ , i.e. a category $\mathcal H$ equipped with a functor $B:\mathcal H\to\mathcal Iso$ . The full subcategories $\mathcal A:=B^{-1}(0)$ and $\mathcal B:=B^{-1}(1)$ are called banks of the bridge, and we write $\mathcal H:\mathcal A \rightleftharpoons \mathcal B$ .
The arrows in the $B$ -preimage of the two arrows in $\mathcal Iso$ are referred to as through arrows or heteromorphisms in the bridge.
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