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Disclaimer: I know practically nothing about category theory, so sorry if the answer is "obvious".

Consider a category C, a subcategory D and a "closure" (i.e. idempotent) functor F : C → D. Then, given an object X ∈ D, what is the "smallest" Y ∈ C such that F(Y) = X?

It's not clear that "smallest" is well-defined for all such categories and idempotent functors, but here are two motivational examples from algebra where the categories are concrete (so objects can be ordered by isomorphic inclusion) and the functors are a form of set extension:

C the category of fields, D the category of algebraically closed fields, F algebraic closure.

C the category of (integral) domains, D the category of fields, F "field of fractions".

There are more examples of the sort of "closure functors" I'm talking about in the answers to this question.

Edit: a followup question - if the smallest such Y is X itself, i.e. X has no proper subobjects Y ∈ C such that F(Y) = X, what does this imply about X?

Disclaimer: I know practically nothing about category theory, so sorry if the answer is "obvious".

Consider a category C, a subcategory D and a "closure" (i.e. idempotent) functor F : C → D. Then, given an object X ∈ D, what is the "smallest" Y ∈ C such that F(Y) = X?

It's not clear that "smallest" is well-defined for all such categories and idempotent functors, but here are two motivational examples from algebra where the categories are concrete (so objects can be ordered by isomorphic inclusion) and the functors are a form of set extension:

C the category of fields, D the category of algebraically closed fields, F algebraic closure.

C the category of (integral) domains, D the category of fields, F "field of fractions".

There are more examples of the sort of "closure functors" I'm talking about in the answers to this question.

Edit: a followup question - if the smallest such Y is X itself, i.e. X has no proper subobjects Y ∈ C such that F(Y) = X, what does this imply about X?

Disclaimer: I know practically nothing about category theory, so sorry if the answer is "obvious".

Consider a category C, a subcategory D and a "closure" (i.e. idempotent) functor F : C → D. Then, given an object X ∈ D, what is the "smallest" Y ∈ C such that F(Y) = X?

It's not clear that "smallest" is well-defined for all such categories and idempotent functors, but here are two motivational examples from algebra where the categories are concrete (so objects can be ordered by isomorphic inclusion) and the functors are a form of set extension:

C the category of fields, D the category of algebraically closed fields, F algebraic closure.

C the category of (integral) domains, D the category of fields, F "field of fractions".

There are more examples of the sort of "closure functors" I'm talking about in the answers to this question.

Disclaimer: I know practically nothing about category theory, so sorry if the answer is "obvious".

Consider a category C, a subcategory D and a "closure" (i.e. idempotent) functor F : C → D. Then, given an object X ∈ D, what is the "smallest" Y ∈ C such that F(Y) = X?

It's not clear that "smallest" is well-defined for all such categories and idempotent functors, but here are two motivational examples from algebra where the categories are concrete (so objects can be ordered by isomorphic inclusion) and the functors are a form of set extension:

C the category of fields, D the category of algebraically closed fields, F algebraic closure.

C the category of (integral) domains, D the category of fields, F "field of fractions".

There are more examples of the sort of "closure functors" I'm talking about in the answers to this question.

Edit: a followup question - if the smallest such Y is X itself, i.e. X has no proper subobjects Y ∈ C such that F(Y) = X, what does this imply about X?

Disclaimer: I know practically nothing about category theory, so sorry if the answer is "obvious".

Consider a category C, a subcategory D and a "closure" (i.e. idempotent) functor F : C → D. Then, given an object X ∈ D, what is the "smallest" Y ∈ C such that F(Y) = X?

It's not clear that "smallest" is well-defined for all such categories and idempotent functors, but here are two motivational examples from algebra where the categories are concrete and the functors are a form of set extension:

C the category of fields, D the category of algebraically closed fields, F algebraic closure.

C the category of (integral) domains, D the category of fields, F "field of fractions".

There are more examples of the sort of "closure functors" I'm talking about in the answers to this question.

Disclaimer: I know practically nothing about category theory, so sorry if the answer is "obvious".

Consider a category C, a subcategory D and a "closure" (i.e. idempotent) functor F : C → D. Then, given an object X ∈ D, what is the "smallest" Y ∈ C such that F(Y) = X?

It's not clear that "smallest" is well-defined for all such categories and idempotent functors, but here are two motivational examples from algebra where the categories are concrete (so objects can be ordered by isomorphic inclusion) and the functors are a form of set extension:

C the category of fields, D the category of algebraically closed fields, F algebraic closure.

C the category of (integral) domains, D the category of fields, F "field of fractions".

There are more examples of the sort of "closure functors" I'm talking about in the answers to this question.