double category | Physics Library

0.1 Background

Charles Ehresmann defined in 1963 a double category 𝒟 as an internal category in the category of small categories Cat.

0.2 Double category definition

Definition 0.1. A double category 𝒟 consists of:

a set of objects,
a set of horizontal morphisms $f : A → B,$
a set of vertical morphisms $j : A → C,$
and
a class of squares with source and target as shown in the following diagrams: $f A|----- B | |g k C -----D h$

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

i. Horizontal: $f1 f2 A − → B −→ C = [f1,f2] = f2 ∘ f1$

$h h A 1−A→ A −f1→ B = A −f1→ B = A −f→1 B −1B→ B$
ii. Vertical: $j1 j2 [A − → B −→ C ]vert = [j1,j2]vert.= j2 ∘ j1$

$v v 1A j1 j1 j1 1B [A −→ A − → B = A −→ B = A − → B − → B ]vert.$
Compositions for square diagrams in a double category 𝒟:
iii. Horizontal composition: $f1 f2 [f1f2] A -----B| [α]“ ∘ ′′ B -----C| [β] = A|-----C| [αβ]. | | | | | | j k k l j l D --g--E E -g---F D -g-g-F 1 2 12$
iv. Vertical composition of squares in 𝒟: [αβ]_vert. is expressed as $A --f--B [αβ] . | | v [j1j2]v| [k1k2]v ----- E h F$

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

-−-−- [α] [β] = [[αβ ] over [γ δ]]vert.= [αγ ]v ∘ [βδ ]v. || | [γ]----- [δ] − −

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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