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Left adjoints preserve every colimit that exists
Statement
If is a left adjoint and a diagram has a colimit, then applying to a colimiting cocone produces a colimit of . Thus left adjoints preserve every colimit that exists.
Facts & Assumptions
Given: An adjunction and a diagram in with a colimit.
A formal theorem derived from the category axioms has a formal dual obtained by reversing morphisms and exchanging each notion with its opposite-category version (Every theorem about categories has a formal dual obtained by reversing morphisms and composition).
A cocone is colimiting in a category exactly when the reversed family is a limiting cone in the opposite category (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).
Right adjoints preserve every limit that exists (Right adjoints preserve every limit that exists).
Proof
Passing to opposite categories turns into , so is a right adjoint.
By [F2], the given colimit in is a limit in .
Apply [L1] to : the reversed image cone is limiting in .
Translating back with [F2], the image cocone under is colimiting in .
Depends on
Used by
- A right adjoint is left exact and a left adjoint is right exact Corollary
- A right adjoint preserves ends and a left adjoint preserves coends Corollary
- In a cartesian closed category, any initial object is strict Corollary
- Right adjoints preserve monomorphisms and left adjoints preserve epimorphisms Corollary
- Every functor with a left adjoint also has a right adjoint False statement
- Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
- Filtered colimits of sheaves and sections over compact opens Lemma
- Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers Proposition
- A reflective subcategory has every ambient colimit, obtained by reflecting an ambient colimit Theorem
- In a biclosed monoidal category tensor is cocontinuous in each variable Theorem
- Tensor product in a multitensor category is biexact Theorem
- The presheaf category on a small category is the free cocompletion Theorem
Dependency tree · two levels
7 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
- Emily Riehl, Category Theory in Context, 2nd ed., Theorem 4.6.2 (standard reference, not scraped)
- Tom Leinster, Basic Category Theory, Section 6.3 (standard reference, not scraped)