double category

Background

Charles Ehresmann defined in 1963 a double category $ \mathcal{D}$ as an internal category in the category of small categories $ \bf {Cat}$ .

Double category definition

Definition 0.1   A double category $ \mathcal{D}$ consists of:

with compositions and units of the double category that satisfy the following axioms:

Moreover, all compositions are associative and unital, and also subject to the Interchange Law:

$\displaystyle \xymatrix{
{[\alpha]}\ar[r]^{--}\ar[d]_{\vert}&{[\beta]}\ar[d]^{\...
... ~~over~~ [\gamma \delta]]}_{vert.} = [\alpha \gamma]_v \circ [\beta \delta]_v.$

Unit morphisms are also subject to the axioms of the double category. For further details on double categories and examples please see the related free download PDF file.



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