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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:

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

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  • $\begingroup$ Somebody was unhappy with my question! I hope the question is more specific after the update. $\endgroup$
    – qartal
    Dec 26, 2014 at 1:24
  • $\begingroup$ You may get some use out of looking at something on Topos theory. Also, out of looking at something on generalized elements. $\endgroup$
    – user14972
    Dec 26, 2014 at 1:34
  • $\begingroup$ @Hurkyl, I am coming from the engineering background, and unfortunately not familiar with Topos theory. I would check the generalized elements concepts though. Thanks. $\endgroup$
    – qartal
    Dec 26, 2014 at 1:39
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    $\begingroup$ I think these things are treated very nicely in Mitchell's classic book. $\endgroup$
    – user158047
    Dec 26, 2014 at 7:20
  • $\begingroup$ Thank @JakobWerner, I see they are discussed in this book as you mentioned. It seems these concepts (i.e., union, intersection and ..) which could be intuitively described in set theory by using Van diagrams, get to be complicated definitions in category theory! $\endgroup$
    – qartal
    Dec 26, 2014 at 19:16

1 Answer 1

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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.

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    $\begingroup$ Thanks @Qiaochu. The way you cast the intersection and union to product and co-products is elegant. How can I study this transformation elaborately with concrete examples? $\endgroup$
    – qartal
    Dec 26, 2014 at 19:24
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    $\begingroup$ I don't know what you're hoping to get out of this, but you could try looking at some topos theory textbooks. In particular it might help to look at how familiar set-theoretic operations are interpreted in topoi other than $\text{Set}$. $\endgroup$ Dec 27, 2014 at 10:58

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