# PlanetPhysics/Double Category

### Background

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

### Double category definition

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

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

• i. Horizontal: $\displaystyle A\buildrel f_1 \over \longrightarrow B \buildrel f_2 \over \longrightarrow C = [f_1, f_2]= f_2 \circ f_1$ $\displaystyle A\buildrel 1^h_A \over \longrightarrow A \buildrel f_1 \over \longrightarrow B = A\buildrel f_1 \over \longrightarrow B = A \buildrel f_1 \over \longrightarrow B \buildrel 1^h_B \over \longrightarrow B$
• ii. Vertical: $\displaystyle [A\buildrel j_1 \over \longrightarrow B \buildrel j_2 \over \longrightarrow C]_{vert} = [j_1, j_2]_{vert.}= j_2 \circ j_1$ $\displaystyle [A\buildrel 1^v_A \over \longrightarrow A \buildrel j_1 \over \longrightarrow B = A\buildrel j_1 \over \longrightarrow B = A \buildrel j_1 \over \longrightarrow B \buildrel 1^v_B \over \longrightarrow B]_{vert.}$ Compositions for \htmladdnormallink{square diagrams {http://planetphysics.us/encyclopedia/Commutativity.html} in a double category ${\displaystyle {\mathcal {D}}}$ :}
• iii. Horizontal composition: $\displaystyle \xymatrix{ {A}\ar[r]^{f_1}\ar[d]_{j}&{B}\ar[d]^{k}\\ {D}\ar[r]_{g_1}&{E}}[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 20:49, 25 June 2015 (UTC)[\alpha]"\circ" \xymatrix{ {B}\ar[r]^{f_2}\ar[d]_{k}&{C}\ar[d]^{l}\\ {E}\ar[r]_{g_2}&{F}}[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 20:49, 25 June 2015 (UTC)[\beta] = \xymatrix{ {A}\ar[r]^{[f_1f_2]}\ar[d]_{j}&{C}\ar[d]^{l}\\ {D}\ar[r]_{g_1g_2}&{F}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 20:49, 25 June 2015 (UTC)[\alpha \beta].$
• iv. Vertical composition of squares in ${\displaystyle {\mathcal {D}}}$ : ${\displaystyle {[\alpha \beta ]}_{vert.}}$  is expressed as $\displaystyle \xymatrix{ {A}\ar[r]^{f}\ar[d]_{[j_1 j_2]_v}&{B}\ar[d]^{[k_1 k_2]_v}\\ {E}\ar[r]_{h}&{F}}[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 20:49, 25 June 2015 (UTC)[\alpha \beta]_v.$

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

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

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.