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May 16, 2013

The Propositional Fracture Theorem

Posted by Mike Shulman

Suppose XX is a topological space and UXU\subseteq X is an open subset, with closed complement K=XUK = X\setminus U. Then UU and KK are, of course, topological spaces in their own right, and we have X=UKX = U\sqcup K as a set. What additional information beyond the topologies of UU and KK is necessary to enable us to recover the topology of XX on their disjoint union?

Recall that the subspace topologies of UU and KK say that for each open VXV\subseteq X, the intersections VUV\cap U and VKV\cap K are open in UU and KK, respectively. Thus, if a subset of XX is to be open, it must yield open subsets of UU and KK when intersected with them. However, this condition is not in general sufficient for a subset of XX to be open — it does define a topology on XX, but it’s the coproduct topology, which may not be the original one.

One way we could start is by asking what sort of structure relating UU and KK we can deduce from the fact that both are embedded in XX. For instance, suppose AUA\subseteq U is open. Then there is some open VXV\subseteq X such that VU=AV\cap U = A. But we could also consider VKV\cap K, and ask whether this defines something interesting as a function of AA.

Of course, it’s not clear that VKV\cap K is a function of AA at all, since it depends on our choice of VV such that VU=AV\cap U = A. Is there a canonical choice of such VV? Well, yes, there’s one obvious canonical choice: since UU is open in XX, AA is also open as a subset of XX, and we have AU=AA\cap U = A. However, AK=A\cap K = \emptyset, so choosing V=AV=A wouldn’t be very interesting.

The choice V=AV=A is the smallest possible VV such that VU=AV\cap U = A. But there’s also a largest such VV, namely the union of all such VV. This set is open in XX, of course, since open sets are closed under arbitrary unions, and since intersections distribute over arbitrary unions, its intersection with UU is still AA.

Let’s call this set i *(A)i_\ast(A). In fact, it’s part of a triple of adjoint functors i !i *i *i_! \dashv i^\ast \dashv i_\ast between the posets O(U)O(U) and O(X)O(X) of open sets in UU and XX, where i *:O(X)O(U)i^\ast:O(X)\to O(U) is defined by i *(V)=VUi^\ast(V) = V\cap U, and i !:O(U)O(X)i_!:O(U)\to O(X) is defined by i !(A)=Ai_!(A)=A. Here ii denotes the continuous inclusion UXU\hookrightarrow X.

Now we can consider the intersection i *(A)Ki_\ast(A) \cap K, which I’ll also denote j *i *(A)j^\ast i_\ast(A), where j:KXj:K\hookrightarrow X is the inclusion. It turns out that this is interesting! Consider the following example, which is easy to visualize:

  • X= 2X = \mathbb{R}^2.
  • U={(x,y)|x<0}U = \{ (x,y) | x \lt 0 \}, the open left half-plane.
  • K={(x,y)|x0}K = \{ (x,y) | x \ge 0 \}, the closed right half-plane.

If an open subset AUA\subseteq U “doesn’t approach the boundary” between UU and KK, such as the open disc of radius 11 centered at (2,0)(-2,0), then it’s fairly easy to see that i *(A)=A{(x,y)|x>0}i_\ast(A) = A \cup \{(x,y) | x \gt 0 \}, and therefore j *i *(A)={(x,y)|x>0}j^\ast i_\ast(A) = \{(x,y) | x \gt 0 \} is the open right half-plane.

On the other hand, consider some open subset AUA\subseteq U which does approach the boundary, such as A={(x,y)|x 2+y 2<1andx<0} A = \{ (x,y) | x^2 + y^2 \lt 1 \;\text{and}\; x \lt 0 \} the intersection with UU of the open disc of radius 11 centered at (0,0)(0,0). A little thought should convince you that in this case, i *(A)i_\ast(A) is the union of the open right half-plane with the whole open disc of radius 11 centered at (0,0)(0,0). Therefore, j *i *(A)j^\ast i_\ast(A) is the open right half-plane together with the strip {(0,y)|1<y<1}\{ (0,y) | -1 \lt y \lt 1 \}.

This example suggests that in general, j *i *(A)j^\ast i_\ast(A) measures how much of the “boundary” between UU and KK is “adjacent” to AA. I leave it to some enterprising reader to try to make that precise. Here’s another nice exercise: what can you say about i *j *(B)i^\ast j_\ast(B) for an open subset BKB\subseteq K?

Let us however go back to our original question of recovering the topology of XX. Suppose AUA\subseteq U and BKB\subseteq K are open such that ABA\cup B is open in XX; how does this latter fact manifest as a property of AA and BB? Note first that (AB)U=A(A\cup B) \cap U = A. Thus, since i *(A)i_\ast(A) is the largest VV such that VU=AV\cap U = A, we have ABi *(A)A\cup B \subseteq i_\ast(A), and therefore B=j *(AB)j *i *(A)B = j^\ast(A\cup B) \subseteq j^\ast i_\ast(A). Let me say that again: Bj *i *(A). B \subseteq j^\ast i_\ast(A). This is a relationship between AA and BB which is expressed purely in terms of the topological spaces UU and KK and the function j *i *:O(U)O(K)j^\ast i_\ast : O(U) \to O(K), which we have just shown is necessary for ABA\cup B to be open in XX.

In fact, it is also sufficient! For suppose this to be true. Since BB is open in KK, there is some open CXC\subseteq X such that CK=BC\cap K = B. Given such a CC, the union CUC\cup U also has this property, since UK=U\cap K = \emptyset. Note that in fact CU=BUC\cup U = B\cup U, and also BU=j *(B)B\cup U = j_\ast (B), the largest open subset of XX whose intersection with KK is BB. (Since KK, unlike UU, is not open, there may not be a smallest such, but there is always a largest such.) Now I claim we have AB=j *(B)i *(A) A \cup B = j_\ast (B) \cap i_\ast(A) To show this, it suffices to show that the two sides become equal after intersecting with UU and with KK. For the first, we have (j *(B)i *(A))U=j *(B)(i *(A)U)=j *(B)A=A=(AB)U (j_\ast (B) \cap i_\ast(A)) \cap U = j_\ast (B) \cap (i_\ast(A) \cap U) = j_\ast (B) \cap A = A = (A\cup B) \cap U and for the second we have (j *(B)i *(A))K=(j *(B)K)i *(A)=Bi *(A)=B=(AB)K (j_\ast (B) \cap i_\ast(A)) \cap K = (j_\ast (B) \cap K) \cap i_\ast(A) = B \cap i_\ast(A) = B = (A\cup B) \cap K using the assumption at the step Bi *(A)=BB \cap i_\ast(A) = B.

In conclusion, the topology of XX is entirely determined by

  • the induced topology of an open subspace UXU\subseteq X,
  • the induced topology on its closed complement K=XUK = X\setminus U, and
  • the induced function j *i *:O(U)O(K)j^\ast i_\ast : O(U) \to O(K).

Specifically, the open subsets of XX are those of the form ABA\cup B — or equivalently, by the above argument, i *(A)j *(B)i_\ast(A) \cap j_\ast(B) — where AUA\subseteq U is open in UU, BKB\subseteq K is open in KK, and Bj *i *(A)B\subseteq j^\ast i_\ast(A).

An obvious question to ask now is, suppose given two arbitrary topological spaces UU and KK and a function f:O(U)O(K)f:O(U)\to O(K); what conditions on ff ensure that we can define a topology on XUKX\coloneqq U\sqcup K in this way, which restricts to the given topologies on UU and KK and induces ff as j *i *j^\ast i_\ast? We may start by asking what properties j *i *j^\ast i_\ast has. Well, it preserves inclusion of open sets (i.e. AAj *i *(A)j *i *(A)A\subseteq A' \Rightarrow j^\ast i_\ast(A) \subseteq j^\ast i_\ast(A')) and also finite intersections (j *i *(AA)=j *i *(A)j *i *(A)j^\ast i_\ast(A\cap A') = j^\ast i_\ast(A) \cap j^\ast i_\ast(A')), including the empty intersection (j *i *(U)=Kj^\ast i_\ast(U) = K). In other words, it is a finite-limit-preserving functor between posets. Perhaps surprisingly, it turns out that this is also sufficient: any finite-limit-preserving f:O(U)O(K)f:O(U) \to O(K) allows us to glue UU and KK in this way; I’ll leave that as an exercise too.

Okay, that was some fun point-set topology. Now let’s categorify it. Open subsets of XX are the same as 0-sheaves on it, i.e. sheaves of truth values, or of subsingleton sets, and the poset O(X)O(X) is the (0,1)-topos of 0-sheaves on XX. So a certain sort of person immediately asks, what about nn-sheaves for n>0n\gt0?

In other words, suppose we have XX, UU, and KK as above; what additional data on the toposes Sh(U)Sh(U) and Sh(K)Sh(K) of sheaves (of sets, or groupoids, or homotopy types, etc.) allows us to recover the topos Sh(X)Sh(X)? As in the posetal case, we have adjunctions i !i *i *i_! \dashv i^\ast \dashv i_\ast and j *j *j^\ast \dashv j_\ast relating these toposes, and we may consider the composite j *i *:Sh(U)Sh(K)j^\ast i_\ast : Sh(U) \to Sh(K).

The corresponding theorem is then that Sh(X)Sh(X) is equivalent to the comma category of Id Sh(K)Id_{Sh(K)} over j *i *j^\ast i_\ast, i.e. the category of triples (A,B,ϕ)(A,B,\phi) where ASh(U)A\in\Sh(U), BSh(K)B\in Sh(K), and ϕ:Bj *i *(A)\phi:B \to j^\ast i^\ast(A). This is true for 1-sheaves, nn-sheaves, \infty-sheaves, etc. Moreover, the condition on a functor f:Sh(U)Sh(K)f:Sh(U) \to Sh(K) ensuring that its comma category is a topos is again precisely that it preserves finite limits. Finally, this all works for arbitrary toposes, not just sheaves on topological spaces. I mentioned in my last post some applications of gluing for non-sheaf toposes (namely, syntactic categories).

One new-looking thing does happen at dimension 1, though, relating to what exactly the equivalence Sh(X)(Id Sh(K)j *i *) Sh(X) \simeq (Id_{Sh(K)} \downarrow j^\ast i_\ast) looks like. The left-to-right direction is easy: we send CSh(X)C\in Sh(X) to (i *C,j *C,ϕ)(i^\ast C, j^\ast C, \phi) where ϕ:j *Cj *i *i *C\phi : j^\ast C \to j^\ast i_\ast i^\ast C is j *j^\ast applied to the unit of the adjunction i *i *i^\ast \dashv i_\ast. But in the other direction, suppose given (A,B,ϕ)(A,B,\phi); how can we reconstruct an object of Sh(X)Sh(X)?

In the case of open subsets, we obtained the corresponding object (an open subset of XX) as ABA\cup B, but now we no longer have an ambient “set of points” in which to take such a union. However, we also had the equivalent characterization of the open subset of XX as i *(A)j *(B)i_\ast(A) \cap j_\ast(B), and in the categorified case we do have objects i *(A)i_\ast(A) and j *(B)j_\ast(B) of Sh(X)Sh(X). We might initially try their cartesian product, but this is obviously wrong because it doesn’t incorporate the additional datum ϕ\phi. It turns out that the right generalization is actually the pullback of j *(ϕ)j_\ast(\phi) and the unit of the adjunction j *j *j^\ast\dashv j_\ast at i *(A)i_\ast(A): C j *(B) j *(ϕ) i *(A) j *j *i *(A) \array{ C & \to & j_\ast(B) \\ \downarrow && \downarrow^{j^\ast(\phi)} \\ i_\ast(A) & \to & j_\ast j^\ast i_\ast(A) } In particular, any object CSh(X)C\in Sh(X) can be recovered from i *Ci^\ast C and j *Cj^\ast C by this pullback: C j *j *C i *i *C j *j *i *i *C \array{ C & \to & j_\ast j^\ast C \\ \downarrow && \downarrow \\ i_\ast i^\ast C & \to & j_\ast j^\ast i_\ast i^\ast C }

Now let’s shift perspective a bit, and ask what all this looks like in the internal language of the topos Sh(X)Sh(X). Inside Sh(X)Sh(X), the subtoposes Sh(U)Sh(U) and Sh(K)Sh(K) are visible through the left-exact idempotent monads i *i *i_\ast i^\ast and j *j *j_\ast j^\ast, whose corresponding reflective subcategories are equivalent to Sh(U)Sh(U) and Sh(K)Sh(K) respectively. In the internal type theory of Sh(X)Sh(X), i *i *i_\ast i^\ast and j *j *j_\ast j^\ast are modalities, which I will denote I UI_U and J UJ_U respectively. Thus, inside Sh(X)Sh(X) we can talk about “sheaves on UU” and “sheaves on KK” by talking about I UI_U-modal and J UJ_U-modal types (or sets).

Moreover, these particular modalities are actually definable in the internal language of Sh(X)Sh(X). Open subsets UXU\subseteq X can be identified with subterminal objects of Sh(X)Sh(X), a.k.a. h-propositions or “truth values” in the internal logic. Thus, UU is such a proposition. Now I UI_U is definable in terms of UU by I U(C)=(UC) I_U(C) = (U\to C) I’m using type-theorists’ notation here, so UCU\to C is the exponential C UC^U in Sh(X)Sh(X). The other modality J UJ_U is also definable internally, though a bit less simply: it’s the following pushout: U×C C U J U(C). \array{ U\times C & \to & C\\ \downarrow & & \downarrow \\ U & \to & J_U(C)}. In homotopy-theoretic language, J U(C)J_U(C) is the join of CC and UU, written U*CU\ast C. And if we identify Sh(U)Sh(U) and Sh(K)Sh(K) with their images under i *i_\ast and j *j_\ast, then the functor j *i *:Sh(U)Sh(K)j^\ast i_\ast : Sh(U) \to Sh(K) is just the modality J UJ_U applied to I UI_U-modal types.

Finally, the fact that Sh(X)Sh(X) is the gluing of Sh(U)Sh(U) with Sh(K)Sh(K) means internally that any type CC can be recovered from I U(C)I_U(C), J U(C)J_U(C), and the induced map J U(C)J U(I U(C))J_U(C) \to J_U(I_U(C)) as a pullback: C J U(C) I U(C) J U(I U(C)) \array{ C & \to & J_U(C) \\ \downarrow && \downarrow \\ I_U(C) & \to & J_U(I_U(C)) } Now recall that internally, UU is a proposition: something which might be true or false. Logically, I U(C)=(UC)I_U(C) = (U\to C) has a clear meaning: its elements are ways to construct an element of CC under the assumption that UU is true.

The logical meaning of J UJ_U is somewhat murkier, but there is one case in which it is crystal clear. Suppose UU is decidable, i.e. that it is true internally that “UU or not UU”. If the law of excluded middle holds, then all propositions are decidable — but of course, internally to a topos, the LEM may fail to hold in general. If UU is decidable, then we have U+¬U=1U + \neg U = 1, where ¬U=(U0)\neg U = (U\to 0) is its internal complement. It’s a nice exercise to show that under this assumption we have J U(C)=(¬UC)J_U(C) = (\neg U \to C).

In other words, if UU is decidable, then the elements of J U(C)J_U(C) are ways to construct an element of CC under the assumption that UU is false. In the decidable case, we also have J U(I U(C))=1J_U(I_U(C))=1, so that C=I U(C)×J U(C)C = I_U(C) \times J_U(C) — and this is just the usual way to construct an element of CC by case analysis, doing one thing if UU is true and another if it is false.

This suggests that we might regard internal gluing as a “generalized sort of case analysis” which applies even to non-decidable propositions. Instead of ordinary case analysis, where we have to do two things:

  • assuming UU, construct an element of CC; and
  • assuming not UU, construct an element of CC

in the non-decidable case we have to do three things:

  • assuming UU, construct an element of CC;
  • construct an element of the join U*CU\ast C; and
  • check that the two constructions agree in U*(UC)U*(U\to C).

I have no idea whether this sort of generalized case analysis is useful for anything. I kind of suspect it isn’t, since otherwise people would have discovered it, and be using it, and I would have heard about it. But you never know, maybe it has some application. In any case, I find it a neat way to think about gluing.

Let me end with a tantalizing remark (at least, tantalizing to me). People who calculate things in algebraic topology like to work by “localizing” or “completing” their topological spaces at primes, since it makes lots of things simpler. Then they have to try to put this “prime-by-prime” information back together into information about the original space. One important class of tools for this “putting back together” is called fracture theorems. A simple fracture theorem says that if XX is a pp-local space (meaning that all primes other than pp are inverted) and some technical conditions hold, then there is a pullback square: X X p X (X p ) \array{ X & \to & X^{\wedge}_p\\ \downarrow & & \downarrow\\ X_{\mathbb{Q}} & \to & (X^{\wedge}_p)_{\mathbb{Q}} } where () p (-)^{\wedge}_p denotes pp-completion and () (-)_{\mathbb{Q}} denotes “rationalization” (inverting all primes). A similar theorem applies to any space XX (with technical conditions), yielding a pullback square X pX (p) X ( pX (p)) \array{ X & \to & \prod_p X_{(p)}\\ \downarrow & & \downarrow \\ X_{\mathbb{Q}} & \to & \Big(\prod_p X_{(p)}\Big)_{\mathbb{Q}} } where () (p)(-)_{(p)} denotes localization at pp.

Clearly, there is a formal resemblance to the pullback square involved in the gluing theorem. At this point I feel like I should be saying something about Spec()Spec(\mathbb{Z}). Unfortunately, I don’t know what to say! Maybe some passing expert will enlighten us.

Posted at May 16, 2013 7:47 PM UTC

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3 Comments & 0 Trackbacks

Re: The Propositional Fracture Theorem

I have been wondering about this exact thing recently, though not in quite so much detail! This post gave me some new ideas that I will have to sleep on, but hopefully someone can come by and enlighten us both.

Posted by: Patrick Durkin on May 17, 2013 4:33 AM | Permalink | Reply to this

Re: The Propositional Fracture Theorem

Because you mention join, and because there is a natural pairing U×(UC)CU \times (U\to C) \to C, there is also a Hopf construct U(UC)ΣCU\star (U \to C) \to \Sigma C, where we might as well have ΣC:=boolC\Sigma C := bool \star C ; with you, at the momement, in that I don’t know if it’s good for anything, but it seemed worth mentioning anyway. Of course, this is supposing that what you mean by “join” is something close to what I mean by “join”, though perhaps the coincidence is invariant under semantics of join?

Posted by: Jesse McKeown on May 17, 2013 5:01 AM | Permalink | Reply to this

Re: The Propositional Fracture Theorem

I mean the same thing by join as a topologist does: the homotopy pushout of two spaces under their cartesian product.

Posted by: Mike Shulman on May 21, 2013 9:08 PM | Permalink | Reply to this

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