TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Evaluation is the colimit over the slice category

Statement

Let $F : C \to E$ be a functor and let $c$ be an object of $C$ . Then the diagram

$C / c ⟶ C \overset{F}{⟶} E$

has a colimit, and that colimit is $F (c)$ .

The proof is again direct: the identity arrow $1_{c} : c \to c$ is terminal in the slice category.

Facts & Assumptions

Given: A functor $F : C \to E$ and an object $c$ of $C$ .

[F1]

The slice category $C / c$ has objects arrows $u : d \to c$ , and a morphism from $u : d \to c$ to $u^{'} : d^{'} \to c$ is an arrow $a : d \to d^{'}$ with $u^{'} \circ a = u$ (Comma category, slice category, and coslice category).

[F2]

A colimit is an initial cocone (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[L1]

The comma-category formulas compute left Kan extensions (Comma-category limit and colimit formulae compute Kan extensions).

Proof

technique · direct

1.1F1L1

Specialize the left-Kan part of [L1] to $K = 1_{C}$ . Then the indexing category $(1_{C} ↓ c)$ is exactly the slice category $C / c$ of [F1].

2.1F1F2step 1.1∎

The object $(c, 1_{c})$ is terminal in $C / c$ : for any object $u : d \to c$ , the required morphism from $u$ to $(c, 1_{c})$ is $u$ itself, and it is unique because its defining equation is $1_{c} \circ a = u$ . Therefore the image of $(c, 1_{c})$ under the diagram is a colimit object. That image is $F (c)$ , and its colimiting cocone has components $F (u) : F (d) \to F (c)$ . By [F2], this says precisely that $F (c)$ is the colimit of the diagram on $C / c$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

E. Riehl, Category Theory in Context, 2nd ed., Proposition 6.5.6 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Proposition 4.7.2 (standard reference, not scraped)