nLab quadrable cospan

Idea

A cospan

$$\array{ F_B &&&& F_C \\ & \searrow && \swarrow \\ && F_A }$$

in a category $\mathcal{C}$ is called quadrable , if there exists a cone

$$\array{ && N \\ & \swarrow && \searrow \\ F_B &&&& F_C \\ & \searrow && \swarrow \\ && F_A } \,.$$

Definition

Let $Sp = \left\{\array{ B &&&& C \\ & \nwarrow && \nearrow \\ && A}\right\}$ denote the span diagram category, that is, the category with three objects $A,B,C$ and two non-identity morphisms $A\to B$ and $A\to C$.

Let $\mathcal{C}$ be any category and let $\Delta$ denote the diagonal $\mathcal{C}\to [Sp^{op},\mathcal{C}]$ into the functor category sending an object $X\mapsto c_X$, where $c_X$ is the constant functor sending all objects and all morphisms of $Sp^{op}$ to $X$ and $id_X$ respectively.

For $F\in [Sp^{op}, \mathcal{C}]$ let $*_F$ denote the unique functor $*\to [Sp^{op},\mathcal{C}]$ from the terminal category such that the unique object of $*$ maps to $F$.

We say that a cospan $F$ in a category $\mathcal{C}$, that is, an object of the functor category $[Sp^{op},\mathcal{C}]$ is quadrable if there exists a cone $N$ for $F$, that is, an object $N$ in the comma category $(\Delta \downarrow *_F)$.

Dually, we say that a span $G$ in a category $C$, that is, an object of the functor category $[Sp,\mathcal{C}]$ is coquadrable if there exists a cocone $N$ for $G$, that is, an object $N$ in the comma category $(*_G \downarrow \Delta)$.

We say that a category $\mathcal{C}$ is quadrable (resp. coquadrable) if all cospans (resp. spans) in $C$ are quadrable (resp. coquadrable).

Note on terminology

The term quadrable is supposed to be a translation of the French carrable , whose use is more wide-spread. (However, note that this is a different meaning than carrable morphism).