ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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The exponential of two small categories computed on a walking-arrow source

Example

Let $I$ be the walking-arrow category with objects $0, 1$ and one nonidentity arrow $u : 0 \to 1$ . For any small category $E$ , a functor $I \to E$ is exactly an arrow of $E$ .

Facts & Assumptions

Given: The walking-arrow category $I$ and a small category $E$ .

[L1]

In $C a t$ , the exponential by $I$ is the functor category $[I, E]$ (The category of small categories is cartesian closed).

Verification

technique · direct

1.1givenalgebra

To give a functor $F : I \to E$ is to choose the two objects $F (0), F (1)$ and the image $F (u) : F (0) \to F (1)$ . Thus the objects of the exponential $E^{I}$ are precisely the arrows of $E$ .

2.1step 1.1algebra

A morphism between two such functors is a natural transformation, so it is exactly a commutative square in $E$ . Therefore $E^{I}$ is the arrow category of $E$ .

3.1step 2.1L1∎

This is the concrete exponential object described by The category of small categories is cartesian closed: functors $C \times I \to E$ curry to functors $C \to E^{I}$ .

Depends on

The category of small categories is cartesian closed

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tom Leinster, Basic Category Theory, Example 6.3.17 (standard reference, not scraped)