Categorical bridge

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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 $\varphi\mid_{\mathbb B}= \mathrm{id}_{\mathbb B}$ .