Charles Ehresmann defined in 1963 a double category
as an internal category
in the category of small categories
.
and
with compositions and units of the double category that satisfy the following axioms:
Compositions for square diagrams in a double category
Moreover, all compositions are associative and unital, and also subject to the Interchange Law:
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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