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Idea

A cospan

in a category $𝒞$ is called quadrable , if there exists a cone

Definition

Let $\mathrm{Sp}=\left\{\begin{array}{ccccc}B& & & & C\\ & \nwarrow & & \nearrow \\ & & A\end{array}\right\}$ 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 $𝒞$ be any category and let $\Delta$ denote the diagonal $𝒞\to [{\mathrm{Sp}}^{\mathrm{op}},𝒞]$ 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 ${\mathrm{Sp}}^{\mathrm{op}}$ to $X$ and ${\mathrm{id}}_X$ respectively.

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

We say that a cospan $F$ in a category $𝒞$, that is, an object of the functor category $[{\mathrm{Sp}}^{\mathrm{op}},𝒞]$ 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 $[\mathrm{Sp},𝒞]$ 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 $𝒞$ 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. It appears for instance in

• Bertrand Toen, Cours de Master 2 : Champs algébriques (2006-2007) cours 2 (web)