Categorical bridge

From Wikipedia, the free encyclopedia

This article or section has multiple issues. Please help improve the article or discuss these issues on the talk page.

It does not cite any references or sources. Please help improve it by citing reliable sources. Tagged since April 2008.
It may contain original research or unverifiable claims. Tagged since April 2008.
It may be too technical for a general audience. Please help make it more accessible. Tagged since April 2008.
It may contain material not appropriate for an encyclopedia. Tagged since April 2008.

In category theory, a discipline in mathematics, a bridge between categories $\mathbb A$ and $\mathbb B$ is a category $\mathbb H$ such that $\mathbb A$ and $\mathbb B$ are disjoint full subcategories of $\mathbb H$ and $\mathrm{Ob}\mathbb H=\mathrm{Ob}\mathbb A\cup \mathrm{Ob}\mathbb B$ . Morphisms of $\mathbb A$ and $\mathbb B$ are called homomorphisms and the rest (passing between $\mathbb A$ and $\mathbb B$ ) are called heteromorphisms.

In notation: $\mathbb H:\mathbb A\leftrightharpoons \mathbb B$ .

As an example, the empty bridge between two categories is just their disjoint union.

A directed bridge from $\mathbb A$ to $\mathbb B$ is a bridge without arrows of the form $B\to A$ (where $B\in\mathrm{Ob}\mathbb B$ and $A\in\mathrm{Ob}\mathbb A$ ). We can easily see that directed bridges and profunctors (i.e. functors $F:\mathbb A^{op}\times\mathbb B\to\mathrm{Set}$ ) are eventually the same [by identifying $F (A, B)$ with the set of heteromorphisms $A\to B$ ].

[edit] Bridge morphism

A morphism between bridges $\mathbb H, \mathbb K:\mathbb A\leftrightharpoons \mathbb B$ is just a functor $\varphi:\mathbb H\to\mathbb K$ which is identical on both $\mathbb A$ and $\mathbb B$ , i.e. $\varphi\mid_{\mathbb A}= \mathrm{id}_{\mathbb A}$ and $\varphi\mid_{\mathbb B}= \mathrm{id}_{\mathbb B}$ .