Concept

Subobject

Summary
In , a branch of mathematics, a subobject is, roughly speaking, an that sits inside another object in the same . The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements. The concept to a subobject is a . This generalizes concepts such as quotient sets, quotient groups, quotient spaces, quotient graphs, etc. Definitions An appropriate categorical definition of "subobject" may vary with context, depending on the goal. One common definition is as follows. In detail, let A be an object of some category. Given two monomorphisms :u: S \to A \ \text{and} \ v: T\to A with codomain A, we define an equivalence relation by u \equiv v if
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