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

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

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 $sSet$-category $S : \Delta \to sSet Cat$ that induces the homotopy coherent nerve. See there for more details on how this encodes the exchange laws.

See also

• Eckmann-Hilton argument

• strict 2-category