Categorical equivalents of Set theory concepts

Question

Update: I am updating my question to be more precise.

I am studying Category of finite sets and functions (FinSet). I am aware that for some of the concepts in Set Theory there are well-known categorical counterparts:

• Cartesian Product ($A \times B$) | Product
• Disjoint Union ($ A+B $) | Co-product
• Subseting ($A \subset B$)| subObject
• PowerSet ($P(A)$) | Exponential object

I am looking to complete this list and looking to see how the following concepts are interpreted in categorical sense.

• Union ($A \cup B$) | ?
• Intersection ($A \cap B$) | ?
• Membership ($A \in B$) | ?
• Cardinality ($|A|$) | ?
• subtraction ($A - B$) | ?
• Complement ($A'$) | ?

Cheers

Answer 1

As far as category theory is concerned, there is no such thing as the non-disjoint union or intersection of two sets, and similarly there is no such thing as the membership relation among sets. What there is is the union or intersection of two subsets of a set. Categorically, subsets $S$ of a set $X$ correspond to (equivalence classes of) monomorphisms $S \to X$. The collection of all such monomorphisms forms a category, and the categorical product and coproduct in this category is intersection and union respectively (again, of subsets, not of sets). Intersection in particular is a special kind of pullback.

The cardinality of a set corresponds to its isomorphism class as an object in the category of sets. Subtraction can be described as follows: if $S \to X$ is a monomorphism, then the complement of $S$ is, if it exists, a monomorphism $S' \to X$ which is universal with respect to the property of being disjoint from $S \to X$ in the sense that their intersection is trivial. In more general categories there's no reason this definition should behave particularly nicely.