# Exchange structure

## Definitions

Let ${\mathrm{๐}}_{1}$, ${\mathrm{๐}}_{2}$ be two categories with the same objects. Let $\mathrm{๐}$ be a 2-category. Let ${F}_{1}$ $:{\mathrm{๐}}_{1}โ\mathrm{๐}$ and ${F}_{2}:{\mathrm{๐}}_{2}โ\mathrm{๐}$ be 2-functors such that ${F}_{1}\left(x\right)={F}_{2}\left(x\right)=F\left(x\right)$ for all objects in ${\mathrm{๐}}_{1}$ and ${\mathrm{๐}}_{2}$. Let $\mathrm{โฐ}$ be a fixed class of commutative squares that are preserved under horizontal and vertical composition of the form:

such that:

• $X,Xโฒ,Y,Yโฒ$ are objects in ${\mathrm{๐}}_{1}$ and so are clearly in ${\mathrm{๐}}_{2}$.
• $g$ and $gโฒ$ are arrows of ${\mathrm{๐}}_{1}$
• $f$ and $fโฒ$ are arrows of ${\mathrm{๐}}_{2}$.

Commutative squares satisfying these properties are called mixed squares.

An exchange structure with respect to $\mathrm{โฐ}$ on a pair $\left({F}_{1},{F}_{2}\right)$ for all mixed squares $\left(C\right)$ in $\mathrm{โฐ}$ is a 2-morphism e(C) of $\mathrm{๐}$ (called a 2-morphism associated with the exchange of mixed $\left(C\right)$):

The direction of the 2-morphisms is constant (that is independent of the mixed square). This family of 2-morphisms must satisfy the following conditions:

• They must be compatible with the horizontal composition of mixed squares, that is for every horizontal composition of mixed squares ${\mathrm{๐}}_{1}$, ${\mathrm{๐}}_{1}$:

the following solid arrow diagram commutes:

• They must be compatible with the vertical composition of mixed squares, that is for every vertical composition of mixed squares $\mathrm{๐}$, $\mathrm{๐}โฒ$:

the following solid arrow diagram commutes:

The exchange on $\left({F}_{1},{F}_{2}\right)$ by this family of exchanges on 2-morphisms is sometimes denoted as $\left(e\left(C\right){\right)}_{Cโ\mathrm{โฐ}}$.

## References

J. Ayoub, Les six opรฉrations de Grothendieck et le formalisme des cycles รฉvanescents dans le monde motivique. I., Astรฉrisque No. 314 (2007), x+466 pp. (2008) MR2009h:14032; II. Astรฉrisque No. 315 (2007), vi+364 pp. (2008) MR2009m:14007; also a file at K-theory archive THESE.pdf

Revised on October 14, 2012 07:52:13 by Toby Bartels (69.171.187.28)