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The subobjects of an object in an abelian category form a lattice
Statement
For every object of an abelian category, the subobjects of form a bounded lattice. The meet is pullback, the join is the image construction of The join of two subobjects in an abelian category, the bottom element is the zero subobject, and the top element is .
Facts & Assumptions
Given: An object in an abelian category.
Any two subobjects of admit a least upper bound (The join of two subobjects is their least upper bound).
Any two subobjects of admit a greatest lower bound (The meet of two subobjects is their pullback).
Subobjects are mutual-factorization classes of monomorphisms into (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms).
An abelian category has a zero object, hence zero morphisms (Abelian category).
Proof
By [L1] and [L2], every pair of subobjects of has a join and a meet. That gives the binary lattice operations.
By [L4], the unique map exists. It is monic because every two maps into are equal, so by [L3] it represents a subobject of . Every monomorphism into factors through , and factors through every monomorphism into , so these classes are respectively the top and bottom elements.
Steps 1.1 and 1.2 prove that the subobjects of form a bounded lattice.