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subobject

Terms vs Subobject - What's the difference?

terms | subobject |


As nouns the difference between terms and subobject

is that terms is while subobject is (category theory) given an object b'', a ''subobject'' of it is an equivalence class of objects a_i which relate to ''b'' through monomorphisms m_i : a_i \rightarrow b (if a pair of monomorphisms with codomain ''b'' , which is a monomorphism) of another set.

Isomorphic vs Subobject - What's the difference?

isomorphic | subobject |


As an adjective isomorphic

is related by an isomorphism; having a structure-preserving one-to-one correspondence.

As a noun subobject is

given an object B, a subobject of it is an equivalence class of objects A_i which relate to B through monomorphisms m_i : A_i \rightarrow B. (If a pair of monomorphisms with codomain B factor through each other, then their domains are isomorphic and thus belong to an equivalence class which defines a subobject of B). The subobject generalizes its interpretation in category Set as a set which is a subset (though an inclusion map, which is a monomorphism) of another set.

Codomain vs Subobject - What's the difference?

codomain | subobject |


As nouns the difference between codomain and subobject

is that codomain is (mathematics) the target space into which a function maps elements of its domain it always contains the range of the function, but can be larger than the range if the function is not surjective while subobject is (category theory) given an object b'', a ''subobject'' of it is an equivalence class of objects a_i which relate to ''b'' through monomorphisms m_i : a_i \rightarrow b (if a pair of monomorphisms with codomain ''b'' , which is a monomorphism) of another set.

Monomorphism vs Subobject - What's the difference?

monomorphism | subobject |


In category theory terms the difference between monomorphism and subobject

is that monomorphism is a morphism n such that for any other morphisms f and g, if n \circ f = n \circ g then f = g while subobject is given an object B, a subobject of it is an equivalence class of objects A_i which relate to B through monomorphisms m_i : A_i \rightarrow B. (If a pair of monomorphisms with codomain B factor through each other, then their domains are isomorphic and thus belong to an equivalence class which defines a subobject of B). The subobject generalizes its interpretation in category Set as a set which is a subset (though an inclusion map, which is a monomorphism) of another set.

As nouns the difference between monomorphism and subobject

is that monomorphism is an injective homomorphism while subobject is given an object B, a subobject of it is an equivalence class of objects A_i which relate to B through monomorphisms m_i : A_i \rightarrow B. (If a pair of monomorphisms with codomain B factor through each other, then their domains are isomorphic and thus belong to an equivalence class which defines a subobject of B). The subobject generalizes its interpretation in category Set as a set which is a subset (though an inclusion map, which is a monomorphism) of another set.