nLab retrocell

Definition

In a double category $\mathbb{D}$ with (chosen) companions, a retrocell with boundary is defined to be a cell in $\mathbb{D}$ as follows, where $f^{\ast}$ and $k^{\ast}$ are the (chosen) companions of $f$ and $k$, respectively.

Example

• The double category $\mathbb{S}\mathrm{pan}(\mathcal{C})$ of spans in a category $\mathcal{C}$ has companions. The companion of a morphism $f \colon A \rightarrow B$ is a span $A \overset{1_{A}}{\leftarrow} A \overset{f}{\rightarrow} B$. A retrocell $\varphi$ in $\mathbb{S}\mathrm{pan}(\mathcal{C})$, as denoted above, corresponds to a morphism of spans as follows.

• Retrocells may be used in the definition of monad retromorphism.

References

• Robert Paré, Retrocells, Theory and Applications of Categories, Vol. 40, 2024, No. 5, pp 130-179. [TAC, arXiv:2306.06436]