InternalCategoryInCat

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:

a set of objects,
a set of horizontal morphisms

$\displaystyle f: A \to B,$
a set of vertical morphisms

$\displaystyle j: A \to C,$
and
a class of squares with source and target as shown in the following diagrams:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\ {C}\ar[r]_{h}&{D} } } \end{xy}$

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

i. Horizontal:

$\displaystyle A\buildrel f_1 \over \longrightarrow B \buildrel f_2 \over \longrightarrow C = [f_1, f_2]= f_2 \circ f_1$

$\displaystyle A\buildrel 1^h_A \over \longrightarrow A \buildrel f_1 \over \lon... ... \buildrel f_1 \over \longrightarrow B \buildrel 1^h_B \over \longrightarrow B$
ii. Vertical:

$\displaystyle [A\buildrel j_1 \over \longrightarrow B \buildrel j_2 \over \longrightarrow C]_{vert} = [j_1, j_2]_{vert.}= j_2 \circ j_1$

$\displaystyle [A\buildrel 1^v_A \over \longrightarrow A \buildrel j_1 \over \lo... ...l j_1 \over \longrightarrow B \buildrel 1^v_B \over \longrightarrow B]_{vert.}$
Compositions for square diagrams in a double category $\mathcal{D}$ :
iii. Horizontal composition:

$\displaystyle \xymatrix{ {A}\ar[r]^{f_1}\ar[d]_{j}&{B}\ar[d]^{k}\ {D}\ar[r]_{... ...f_1f_2]}\ar[d]_{j}&{C}\ar[d]^{l}\ {D}\ar[r]_{g_1g_2}&{F}} ~~~~[\alpha \beta].$
iv. Vertical composition of squares in $\mathcal{D}$ : ${[\alpha \beta]}_{vert.}$ is expressed as

$\displaystyle \xymatrix{ {A}\ar[r]^{f}\ar[d]_{[j_1 j_2]_v}&{B}\ar[d]^{[k_1 k_2]_v}\ {E}\ar[r]_{h}&{F}}~~~~[\alpha \beta]_v.$

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.

Background

Double category definition