November 2020 – Profunctors

A category $\ct H$ is a bridge between categories $\ct A$ and $\ct B$ if these are disjoint full subcategories of $\ct H$ and $\ct H$ has no more objects. In notation: $\ct H:\ct A \rightleftharpoons \ct B$ .

In other words, if $\ct Iso$ denotes the category of 2 objects (name them $0,1$ ) and 2 nonidentity arrows, ${\bf i}:0\to 1,\ {\bf j}:1\to 0$ which are inverses of each other, then a bridge is a category over $\ct Iso$ , i.e. a category $\ct H$ equipped with a functor $H:\ct H\to\ct Iso$ . The full subcategories $\ct A:=H^{-1}(0)$ and $\ct B:=H^{-1}(1)$ are called banks of the bridge.
The arrows in the $H$ -preimage of the two arrows ${\bf i}$ and ${\bf j}$ in $\ct Iso$ are referred to as through arrows or heteromorphisms in the bridge.

Month: November 2020

A Bridge Construction