# Contents

## Idea

$\begin{array}{ccccc}{F}_{B}& & & & {F}_{C}\\ & \beta & & \beta \\ & & {F}_{A}\end{array}$\array{ F_B &&&& F_C \\ & \searrow && \swarrow \\ && F_A }

in a category $\mathrm{\pi }$ is called quadrable , if there exists a cone

$\begin{array}{ccc}& & N\\ & \beta & & \beta \\ {F}_{B}& & & & {F}_{C}\\ & \beta & & \beta \\ & & {F}_{A}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && N \\ & \swarrow && \searrow \\ F_B &&&& F_C \\ & \searrow && \swarrow \\ && F_A } \,.

## Definition

Let $\mathrm{Sp}=\left\{\begin{array}{ccccc}B& & & & C\\ & \beta & & \beta \\ & & A\end{array}\right\}$ denote the span diagram category, that is, the category with three objects $A,B,C$ and two non-identity morphisms $A\beta B$ and $A\beta C$.

Let $\mathrm{\pi }$ be any category and let $\mathrm{\Xi }$ denote the diagonal $\mathrm{\pi }\beta \left[{\mathrm{Sp}}^{\mathrm{op}},\mathrm{\pi }\right]$ into the functor category sending an object $X\beta ¦{c}_{X}$, 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 $F\beta \left[{\mathrm{Sp}}^{\mathrm{op}},\mathrm{\pi }\right]$ let ${*}_{F}$ denote the unique functor $*\beta \left[{\mathrm{Sp}}^{\mathrm{op}},\mathrm{\pi }\right]$ from the terminal category such that the unique object of $*$ maps to $F$.

We say that a cospan $F$ in a category $\mathrm{\pi }$, that is, an object of the functor category $\left[{\mathrm{Sp}}^{\mathrm{op}},\mathrm{\pi }\right]$ is quadrable if there exists a cone $N$ for $F$, that is, an object $N$ in the comma category $\left(\mathrm{\Xi }\beta {*}_{F}\right)$.

Dually, we say that a span $G$ in a category $C$, that is, an object of the functor category $\left[\mathrm{Sp},\mathrm{\pi }\right]$ is coquadrable if there exists a cocone $N$ for $G$, that is, an object $N$ in the comma category $\left({*}_{G}\beta \mathrm{\Xi }\right)$.

We say that a category $\mathrm{\pi }$ 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 algΓ©briques (2006-2007) cours 2 (web)

Revised on April 28, 2010 11:31:29 by Harry (67.194.132.91)