nLab exchange law

Context

Higher category theory

higher category theory

Contents

Idea

In higher category theory, the exchange law, or interchange law, states that the multiple ways of forming the composite of a pasting diagram of k-morphisms are equivalent.

Examples

The first exchange law (often called the exchange law) asserts that for composition of 2-morphisms we have an equivalence

$\array{ && 1 &&&& 3 \\ & \nearrow &\Uparrow& \searrow && \nearrow && \searrow \\ 0 &&\to&& 2 && && 4 \\ && &&&& 3 \\ & && && \nearrow &\Uparrow& \searrow \\ 0 &&\to&& 2 && \to && 4 } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ && 1 &&&& 3 \\ & \nearrow && \searrow && \nearrow &\Uparrow& \searrow \\ 0 &&&& 2 && \to && 4 \\ && 1 &&&& \\ & \nearrow &\Uparrow& \searrow && && \\ 0 &&\to && 2 && \to && 4 }$

asserting a compatibility of horizontal composition and vertical composition of 2-morphisms.

In a bicategory this equivalence is an identity. In even higher (and non-semi-strict) category theory, the interchange law becomes a higher morphism itself: the exchanger.

Combinatorics of exchange laws

One way to capture all exchange laws combinatorially is encoded by the cosimplicial $sSet$-cateory $S : \Delta \to sSet Cat$ that induces the homotopy coherent nerve. See there for more details on how this encodes the exchange laws.

Revised on August 31, 2010 04:53:16 by Urs Schreiber (87.212.203.135)