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Filtered colimits commute with finite limits in Set
Statement
For every small filtered category and finite category , and every , the canonical comparison
is a bijection. This includes the empty finite limit.
Facts & Assumptions
Given: The categories and diagram in the statement.
Filteredness combines finitely many objects at a common later stage and coequalizes finitely many parallel arrows (Filtered categories and filtered colimits).
Equality of two elements in a filtered Set-colimit occurs at one common later stage (Two representatives in a filtered colimit of sets are equal exactly when they become equal at one common later stage).
Set-limits are compatible tuples and Set-colimits are tagged quotients (Set has all small limits, realized as compatible tuples in a set-indexed product, Set has all small colimits, realized as a quotient of a set-indexed disjoint union).
Finite limits can be tested using finite products and equalizers (Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals).
A canonical comparison is invertible exactly when the limit is preserved (A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits).
Proof
By [L3], it suffices to prove that a filtered colimit preserves finite products and equalizers. For a finite product, a tuple on the right has finitely many coordinates, each represented at some stage. Repeated use of [F1] moves all representatives to one common stage, producing a tuple there. Thus the comparison is surjective.
If two common-stage tuples have the same image, [L1] makes each coordinate equal at some later stage. There are finitely many coordinates, so repeated use of [F1] moves all those equalities to one stage. The tuples then agree, which proves injectivity.
For an equalizer, an element on the right is represented by at some stage and its two images become equal in the filtered colimit. By [L1], after moving to one later stage its two images are equal there, so it is represented by a stagewise equalizer element. This proves surjectivity.
If two stagewise equalizer elements become equal in the ambient filtered colimit, [L1] makes them equal at a common later stage; functoriality keeps them inside the later equalizer. This proves injectivity.
For the empty product, each stage and the target are singletons. The filtered category is nonempty by [F1], so the colimit of the constant singleton diagram is a singleton, not empty.
Steps 1.1 to 1.5 make every finite-limit comparison bijective. By [L4], filtered colimits preserve finite limits, proving the displayed assertion.
Depends on
- Filtered categories and filtered colimits
- Two representatives in a filtered colimit of sets are equal exactly when they become equal at one common later stage
- Set has all small limits, realized as compatible tuples in a set-indexed product
- Set has all small colimits, realized as a quotient of a set-indexed disjoint union
- Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals
- A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits
Used by
Dependency tree · two levels
22 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, Theorem 3.8.9 (standard reference, not scraped)