nLab exchange structure

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Definitions

Let $\mathcal{C}_{1}$, $\mathcal{C}_2$ be two categories with the same objects. Let $\mathcal{D}$ be a 2-category. Let $F_{1}$ $\colon \mathcal{C}_{1} \rightarrow \mathcal{D}$ and $F_{2}\colon \mathcal{C}_{2} \rightarrow \mathcal{D}$ be 2-functors such that $F_{1}(x) = F_{2}(x) = F(x)$ for all objects in $\mathcal{C}_1$ and $\mathcal{C}_2$. Let $\mathcal{E}$ be a fixed class of commutative squares that are preserved under horizontal and vertical composition of the form:

such that: * $X, X', Y, Y'$ are objects in $\mathcal{C}_1$ and so are clearly in $\mathcal{C}_2$. * $g$ and $g'$ are arrows of $\mathcal{C}_1$ * $f$ and $f'$ are arrows of $\mathcal{C}_2$.

Commutative squares satisfying these properties are called mixed squares.

An exchange structure with respect to $\mathcal{E}$ on a pair $(F_{1},F_{2})$ for all mixed squares $(C)$ in $\mathcal{E}$ is a 2-morphism e(C) of $\mathcal{D}$ (called a 2-morphism associated with the exchange of mixed $(C)$):

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 $\mathcal{C}_1$, $\mathcal{C}_1$:

  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 $\mathcal{C}$, $\mathcal{C}'$:

  the following solid arrow diagram commutes:

The exchange on $(F_{1},F_{2})$ by this family of exchanges on 2-morphisms is sometimes denoted as $(e(C))_{C \in \mathcal{E}}$.

References

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