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My question is the title, really. I am wondering if the intersection of sets can be seen as a categorical construction on the objects of $\mathbf{Set}$.

Boris12
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  • From the view ov category theory, inclusion is not special. Hence the relation among $A,B,A\cap B$ is "indistinguishable" from any other three sets of same cardinality, even if the other sets are in fact disjoint. – Hagen von Eitzen Sep 12 '14 at 09:59

3 Answers3

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$\require{AMScd}$ The more useful category to consider this in is not $\text{Set}$ but the subcategory in which we only consider injective morphisms (inclusions). In that case $A\cap B$ and $A \cup B$, for $A,B \subseteq C$, fit in these pullback and pushout diagrams:

$$ \begin{CD} A\cap B @>>> A \\ @VVV & @VVV \\B @>>> C \end{CD}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{CD} A \cap B @>>> A \\ @VVV & @VVV \\B @>>> A \cup B \end{CD}$$

In $\text{Set}$ one gets the same answers in the special case of injective maps, but pullbacks and pushouts in $\text{Set}$ are not in gneral intersections and unions. To characterize $A\cap B$ and $A\cup B$ as such we need to restrict to the subcategory with injective maps.

The fact that $A\cap B$ and $A\cup B$ are pullback and pushouts are essential to the way Grothendieck topologies generalize topological spaces. One replaces the open sets of a topological space, considered as inclusion maps, with other classes of morphisms with similar formal properties. Intersections of open sets then get replaced with taking fibre products (pullbacks) of morphisms.

Zavosh
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  • Oh...Grothendieck categories are categories of sheaves. And sheaves are presheaves, i.e functors $\mathcal{C}^{\text{op}}\to \mathbf{Set}$. And this is the categorical generalization of sheaves on topological spaces, which are functors $\mathcal{O}(X)^{\text{op}}\to \mathbf{Set}$, where $\mathcal{O}(X)$ is the collection of open sets of a topological space $X$, regarded as a category when we order it by inclusion. And in that category, arrows are injective morphisms. As in our case here. Is this what is hidden behind your last paragraph, or do you mean something else? – Boris12 Sep 12 '14 at 10:55
  • Grothendick topoi, I mean, not categories; in my first sentence. – Boris12 Sep 12 '14 at 11:06
  • I'm referring to Grothendieck topologies (which are not topologies at all). It turns out that you can generalize sheaves even further, by replacing $\mathcal{O}(X)$ by some other category $\mathcal{C}$ associated to $X$. As long as $\mathcal{C}$ has the same formal categorical properties as $\mathcal{O}(X)$, then everything about the theory of sheaves generalizes to functors $\mathcal{C} \rightarrow \text{Set}$. The particular properties $\mathcal{C}$ needs to satisfy have to do with the formal generalization of open sets and open covers. – Zavosh Sep 12 '14 at 11:06
  • For example in the usual definition of a sheaf you need to talk about agreement of sections on 'intersections'. In general you replace those intersections with certain fibre products. – Zavosh Sep 12 '14 at 11:07
  • And yes, once you do this, the category of sheaves 'on $\mathcal{C}$ ' is a Grothendieck topos, and in fact every Grothendieck topos arises this way. This latter fact is a theorem of Giraud. – Zavosh Sep 12 '14 at 11:10
  • @Prometheus The category $\mathcal C$ need not have "the same formal properties as $\mathcal O(X)$". In fact, any small category $\mathcal C$ can be put with the trivial topology (namely, the only sieve on any object is the maximal one) : the resulting Grothendieck topos is then $[\mathcal C^{\mathrm{op}},\mathsf{Set}]$. – Pece Sep 12 '14 at 12:15
  • @Pece: What I mean by formal properties is the axioms of a Grothendieck topology. I don't mean that $\mathcal{C}$ itself actually has to look anything like $\mathcal{O}(X)$. I was trying to avoid particular terminology like sieves and sites, just to get a coherent first impression. – Zavosh Sep 12 '14 at 12:18
  • @Pece: To be clear, what I meant is: As the open sets of a general topological space have formal categorical interpretations, one can generalize topological spaces by asking for a category that satisfies the categorical version of those axioms. Then instead of simply looking at sheaves on the category of open sets $\mathcal{O}(X)$, one can generally look at sheaves on $\mathcal{C}$, and this actually gives a very useful theory. – Zavosh Sep 12 '14 at 12:33
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No, the intersection of two "isolated" sets $A,B$ doesn't have any categorical interpretation. Because for any reasonable meaning of this, we would like to have $A \cap B \cong A' \cap B'$ if $A \cong A'$ and $B \cong B'$. But this is clearly wrong (take $A=B=A'=\{1\}$ and $B'=\{0\}$).

Thus, from the perspective of category theory, the set-theoretic operation $\cap$ doesn't make much sense. What is $\pi \cap \mathbb{R}$ supposed to be? However, it is meaningful to take the intersection with respect to two (injective) maps $A \to C$ and $B \to C$. Namely, then the pullback $A \times_C B$ is the desired intersection. For more on this, see also math.SE/295800 and math.SE/704593 and math.SE/866127.

Community
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Martin Brandenburg
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    In one sentence: Categorical constructions are invariant under isomorphism (isomorphic = undistinguishable by categorical notions), whereas Set-intersection is not invariant under Set-isomorphism (= ordinary bijection). – Thorsten Dec 15 '16 at 15:50
  • @Thorsten: Exactly. – Martin Brandenburg Apr 12 '17 at 11:55
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If $A, B \subseteq C$, $i_1$ is the inclusion from $A$ to $C$ and $i_2$ is the inclusion from $B$ to $C$, then consider the pullback of $i_1$ and $i_2$.

user21929
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    Yes I know; it is the intersection. But...If $\iota:C\to D$ is the inclusion from $C$ to $D$ for some set $D$, consider the pullback of $\iota\circ i_1$ and $\iota\circ i_2$. How is your pullback different to mine? – Boris12 Sep 12 '14 at 10:21
  • "My" pullback and "your" pullback can be constructed by taking $A \cap B$ and the inclusions $A \cap B \subseteq A, B$. – user21929 Sep 12 '14 at 13:06