PlanetPhysics/Double Category

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

Double category definition
A double category $$\mathcal{D}$$ consists of:


 * a set of objects,
 * a set of horizontal morphisms $$f: A \to B,$$
 * a set of vertical morphisms $$j: A \to C,$$ and
 * a class of squares with source and target as shown in the following diagrams: $$\begin{xy} *!C\xybox{ \xymatrix{ {A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\\ {C}\ar[r]_{h}&{D} } }\end{xy}$$

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


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

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

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

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.