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.
Image and coimage in a category with kernels and cokernels
Definition
Let be a morphism in a category with kernels and cokernels (Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers).
The image of is the kernel of a cokernel of :
The coimage of is the cokernel of a kernel of :
This page uses the object notation and . Their defining arrows are the corresponding kernel and cokernel maps.
Depends on
Used by
- The coimage projection is epic and the image inclusion is monic Proposition
- A morphism factors uniquely through its coimage Theorem
- A morphism factors uniquely through its image Theorem
- Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel Theorem
- First isomorphism theorem in an abelian category Theorem
- Freyd's axioms force the additive structure and recover the AB2 definition Theorem
Dependency tree · two levels
3 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
- Gautam Tamme, Algebra II Lecture 9, §9.1 (standard reference, not scraped)
- The Stacks Project, Section 12.3, Definition 12.3.9 (standard reference, not scraped)