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A left Kan extension along a full subcategory inclusion of preorders

Example

Let $A$ be the full subcategory of the chain $0 < 1 < 2$ on the objects $0$ and $1$ , and let $i : A ↪ B$ be the inclusion. Define $F : A \to S e t$ by

$F (0) = \emptyset, F (1) = {*},$

with the unique map $\emptyset \to {*}$ on the unique non-identity arrow.

Then the left Kan extension of $F$ along $i$ exists and is pointwise. It agrees with $F$ on $0$ and $1$ , and its value at the new object $2$ is again the singleton set ${*}$ .

Facts & Assumptions

Given: The preorder categories $A \subseteq B$ and the functor $F$ just described.

[F1]

A preorder may be read as a category with one arrow precisely when the order relation holds, and monotone maps are the functors between them (A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps).

[L1]

The comma-category colimit formula computes the left Kan extension value at an object (Comma-category limit and colimit formulae compute Kan extensions).

[L2]

A pointwise Kan extension along a fully faithful functor restricts back to the original functor by isomorphism (A pointwise Kan extension along a fully faithful functor genuinely extends the original functor, Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

Verification

technique · direct

1.1F1

The inclusion $i$ is fully faithful by [F1]. The comma category $(i ↓ 2)$ has two objects, namely the arrows $0 \to 2$ and $1 \to 2$ , and one non-identity morphism from the first to the second, corresponding to the inequality $0 \leq 1$ in $A$ .

1.2L1

The induced diagram $(i ↓ 2) \to A \overset{F}{\to} S e t$ is therefore just $\emptyset \to {*}$ , whose colimit in $S e t$ is ${*}$ . By [L1], this is the left Kan extension value at $2$ .

2.1L2step 1.2∎

On the objects $0$ and $1$ , the pointwise left Kan extension restricts back to $F$ by [L2]. Hence the left Kan extension along the full subcategory inclusion is the functor sending $0 \mapsto \emptyset$ , $1 \mapsto {*}$ , and $2 \mapsto {*}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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.