Image and coimage in a category with kernels and cokernels

Depends on

• Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers

Used by

• Kernel and image are the inverse and direct images along a morphism Corollary
• Cycle and boundary subobjects of a complex Definition
• Direct and inverse image of a subobject Definition
• Exactness at a node Definition
• The axioms AB5 and AB5* Definition
• The join of two subobjects in an abelian category Definition
• FALSE: the subobject-side definition of exactness needs no canonical image monomorphism False statement
• Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
• Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
• Images and coimages of chain maps are computed degreewise Proposition
• 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
• Exactness is self-dual Theorem
• Finite abelian categories admit finite-dimensional module models Theorem
• First isomorphism theorem in an abelian category Theorem
• Freyd's axioms force the additive structure and recover the AB2 definition Theorem
• Images commute with tensor products in a multitensor category Theorem
• Intrinsic finite category hypotheses give a finite projective generator Theorem
• The arrow-theoretic criterion for exactness 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)