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exchange structure

Exchange structure

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Definitions

Let ๐’ž 1\mathcal{C}_{1}, ๐’ž 2\mathcal{C}_2 be two categories with the same objects. Let ๐’Ÿ\mathcal{D} be a 2-category. Let F 1F_{1} :๐’ž 1โ†’๐’Ÿ\colon \mathcal{C}_{1} \rightarrow \mathcal{D} and F 2:๐’ž 2โ†’๐’ŸF_{2}\colon \mathcal{C}_{2} \rightarrow \mathcal{D} be 2-functors such that F 1(x)=F 2(x)=F(x)F_{1}(x) = F_{2}(x) = F(x) for all objects in ๐’ž 1\mathcal{C}_1 and ๐’ž 2\mathcal{C}_2. Let โ„ฐ\mathcal{E} be a fixed class of commutative squares that are preserved under horizontal and vertical composition of the form:

Layer 1 Yโ€™ Y X Xโ€™ gโ€™ g fโ€™ f

such that:

  • X,Xโ€ฒ,Y,Yโ€ฒX, X', Y, Y' are objects in ๐’ž 1\mathcal{C}_1 and so are clearly in ๐’ž 2\mathcal{C}_2.
  • gg and gโ€ฒg' are arrows of ๐’ž 1\mathcal{C}_1
  • ff and fโ€ฒf' are arrows of ๐’ž 2\mathcal{C}_2.

Commutative squares satisfying these properties are called mixed squares.

An exchange structure with respect to โ„ฐ\mathcal{E} on a pair (F 1,F 2)(F_{1},F_{2}) for all mixed squares (C)(C) in โ„ฐ\mathcal{E} is a 2-morphism e(C) of ๐’Ÿ\mathcal{D} (called a 2-morphism associated with the exchange of mixed (C)(C)):

Layer 1 F(Yโ€™) F(Y) F(X) F(Xโ€™) F(gโ€™) F(g) F(fโ€™) F(f) 1 1 2 2

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 ๐’ž 1\mathcal{C}_1, ๐’ž 1\mathcal{C}_1:

    Layer 1 gโ€™ g f fโ€™ hโ€™ h f

    the following solid arrow diagram commutes:

    Layer 1 F (gโ€™) F (hโ€™) F (hโ€™โˆ˜gโ€™) 1 1 1 F (hโˆ˜g) 1 F (fโ€™) 2 F (f) 2 F (fโ€) 2 F (g) 1 F (h) 1
  • They must be compatible with the vertical composition of mixed squares, that is for every vertical composition of mixed squares ๐’ž\mathcal{C}, ๐’žโ€ฒ\mathcal{C}':

    Layer 1 fโ€™ f eโ€™ e g gโ€™ g

    the following solid arrow diagram commutes:

    Layer 1 F (gโ€) F (gโ€™) F (g) F (e) F (eโ€™) F (fโ€™) F (f) F (eโ€™โˆ˜fโ€™) F (eโˆ˜f) 1 1 1 2 2 2 2 2 2

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

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)