nLab
exchange law

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

$$\begin{array}{ccccccc}& & 1& & & & 3\\ & \nearrow & \Uparrow & \searrow & & \nearrow & & \searrow \\ 0& & \to & & 2& & & & 4\\ & & & & & & 3\\ & & & & & \nearrow & \Uparrow & \searrow \\ 0& & \to & & 2& & \to & & 4\end{array}\simeq \begin{array}{ccccccc}& & 1& & & & 3\\ & \nearrow & & \searrow & & \nearrow & \Uparrow & \searrow \\ 0& & & & 2& & \to & & 4\\ & & 1& & & & \\ & \nearrow & \Uparrow & \searrow & & & & \\ 0& & \to & & 2& & \to & & 4\end{array}$$\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 $\mathrm{sSet}$-cateory $S:\Delta \to \mathrm{sSet}\mathrm{Cat}$ that induces the homotopy coherent nerve. See there for more details on how this encodes the exchange laws.