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The join of two subobjects in an abelian category
Definition
Let and represent two subobjects of an object in an abelian category. Because finite biproducts exist (An abelian category has all finite limits and all finite colimits), there is a unique morphism
whose composites with the two biproduct injections are and .
The join of the two subobjects is the subobject of represented by the image inclusion of in the sense of Image and coimage in a category with kernels and cokernels. It is denoted
The well-definedness obligation on representatives is discharged by The join of two subobjects is their least upper bound ↗.
Depends on
Used by
- Superfluous subobjects and projective covers in an abelian category Definition
- Finite abelian categories admit finite-dimensional module models Theorem
- Second isomorphism theorem in an abelian category Theorem
- The join of two subobjects is their least upper bound Theorem
- The subobject lattice of an abelian category is modular Theorem
- The subobjects of an object in an abelian category form a lattice Theorem
Dependency tree · two levels
9 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
- Daniel Murfet, Abelian Categories, Section 4.2 (standard reference, not scraped)