Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Left exact and right exact functors
Definition
A functor is left exact when it preserves every finite limit that exists in its source category, and right exact when it preserves every finite colimit that exists there. Here finite means indexed by a finite category as in Finite, small, and large limits and colimits; complete and cocomplete categories, and preservation has the meaning of Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors. These terms assert preservation, not existence, of the relevant limits or colimits.
Depends on
Used by
- A right adjoint is left exact and a left adjoint is right exact Corollary
- An acyclic object for a left exact functor Definition
- An acyclic object for a right exact functor Definition
- Exact functor between abelian categories Definition
- G-acyclic object for a left-exact functor Definition
- Right hyperderived functor of a complex Definition
- The horseshoe construction stays short exact after applying a right exact functor Lemma
- A left or right exact functor between abelian categories is automatically additive Theorem
- An equivalence between abelian categories is exact Theorem
- Left exactness, right exactness, and exactness are characterized by short exact sequences Theorem
- The zero-th left derived functor of a right exact functor recovers the functor Theorem
- The zero-th right derived functor of a left exact functor recovers the functor 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., Definition 4.6.7 (standard reference, not scraped)