nLab
quadrability

Contents

Idea

A cospan

F B F C β†˜ ↙ F A \array{ F_B &&&& F_C \\ & \searrow && \swarrow \\ && F_A }

in a category π’ž\mathcal{C} is called quadrable , if there exists a cone

N ↙ β†˜ F B F C β†˜ ↙ F A. \array{ && N \\ & \swarrow && \searrow \\ F_B &&&& F_C \\ & \searrow && \swarrow \\ && F_A } \,.

Definition

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

Let π’ž\mathcal{C} be any category and let Ξ”\Delta denote the diagonal π’žβ†’[Sp op,π’ž]\mathcal{C}\to [Sp^{op},\mathcal{C}] into the functor category sending an object X↦c XX\mapsto c_X, where c Xc_X is the constant functor sending all objects and all morphisms of Sp opSp^{op} to XX and id Xid_X respectively.

For F∈[Sp op,π’ž]F\in [Sp^{op}, \mathcal{C}] let * F*_F denote the unique functor *β†’[Sp op,π’ž]*\to [Sp^{op},\mathcal{C}] from the terminal category such that the unique object of ** maps to FF.

We say that a cospan FF in a category π’ž\mathcal{C}, that is, an object of the functor category [Sp op,π’ž][Sp^{op},\mathcal{C}] is quadrable if there exists a cone NN for FF, that is, an object NN in the comma category (Δ↓* F)(\Delta \downarrow *_F).

Dually, we say that a span GG in a category CC, that is, an object of the functor category [Sp,π’ž][Sp,\mathcal{C}] is coquadrable if there exists a cocone NN for GG, that is, an object NN in the comma category (* G↓Δ)(*_G \downarrow \Delta).

We say that a category π’ž\mathcal{C} is quadrable (resp. coquadrable) if all cospans (resp. spans) in CC 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)