Alphabeta MathHow statement and proof provenance work
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An abelian category that is a preorder is trivial
Statement
If an abelian category is a preorder, then every object is isomorphic to the zero object. In particular it is equivalent to the terminal one-object category.
Facts & Assumptions
Given: An abelian category that is also a preorder.
In a preorder there is at most one morphism between any two fixed objects (Preorder and monotone map).
Abelian categories have a zero object and are balanced (Abelian category, An abelian category is balanced).
Proof
By [L1], any two parallel morphisms in are automatically equal. So every morphism is monic and epic.
Let be the zero object from [L2]. The unique morphisms and are both monic and epic by step 1.1, hence isomorphisms by [L2]. Therefore every object is isomorphic to , and the category is equivalent to the terminal one-object category.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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
- Saunders Mac Lane, Categories for the Working Mathematician, I.1 and VIII.3 (standard reference, not scraped)