nLab exchange law

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

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

1 3 0 2 4 3 0 2 4 1 3 0 2 4 1 0 2 4 \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 sSetsSet-category S:ΔsSetCatS : \Delta \to sSet Cat that induces the homotopy coherent nerve. See there for more details on how this encodes the exchange laws.

See also

Last revised on June 2, 2023 at 04:00:18. See the history of this page for a list of all contributions to it.