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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeAug 28th 2012

    Started work on syntopogenous space.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 28th 2012

    Working on… what? :-)

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeAug 28th 2012

    Is that a question addressed to me?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 28th 2012

    I was just joking… the term “syntopogenous” is one I’d never heard of, and it just sounds so esoteric. Have you known about them a long time?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 28th 2012

    No, I just read about them recently. As far as I know, they aren’t good for anything; I’m just amused that they exist.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 28th 2012

    And that the “underlying topology” functor UnifTopUnif \to Top (and others) can be described as a coreflection.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeAug 28th 2012

    I think I’m done with syntopogenous space now, by the way.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 27th 2014

    Should there be a clause in condition 2 of topogenous relation that ABδCA \cup B \delta C iff AδCA \delta C or BδCB \delta C? (What’s there involves unions only in the second variable.) It seems you’d need this to justify the claim that the opposite of a topogenous relation is also topogenous.

    Is it correct that inserting this clause doesn’t affect any of the results on this page?

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeJun 27th 2014

    Yes, definitely. I'll fix it.

    Is it correct that inserting this clause doesn’t affect any of the results on this page?

    On the contrary, inserting this clause is necessary to make many of them true!

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeJul 18th 2014

    Some changes made, mainly to mention \ll (and \bowtie) earlier and to clarify the role of nontriviality/reflexivity.

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 28th 2014
    • (edited Dec 28th 2014)

    Behind the scenes there has been some further discussion with Victor Porton, including this discussion at MO with an illuminating response from Mamuka Jibladze. This has produced what I regard as (potentially) the most useful way of regarding topogenous relations on a set XX: they can be identified with closed subsets CβX×βXC \hookrightarrow \beta X \times \beta X containing the diagonal, that is to say with reflexive relations from βX\beta X to itself in the pretopos of compact Hausdorff spaces. I have added this remark to syntopogenous space.

    (Briefly: a map PX×PX2P X \times P X \to \mathbf{2} which preserves finite joins in separate arguments amounts to a join-preserving map PXPX2P X \otimes P X \to \mathbf{2} where the tensor product is just that of join-semilattices viewed as (idempotent) commutative monoids. This tensor product gives the coproduct of commutative rigs, and this coproduct is one of Boolean algebras as well. By Stone duality, this corresponds to the product βX×βX\beta X \times \beta X, where the poset of join-preserving maps as above is dual to the poset of ideals in CompHaus(βX×βX,2)CompHaus(\beta X \times \beta X, \mathbf{2}), i.e., is dual to the topology of the Zariski spectrum βX×βX\beta X \times \beta X. This leads to the identification with closed subsets CβX×βXC \subseteq \beta X \times \beta X. Details can be found at my article topogeny.)

    I have not tried to work out what a syntopogeny is from this point of view. But it seems like potentially useful point of view, since so much is known about β\beta.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeDec 29th 2014

    Interesting! How does the correspondence act concretely? E.g. if δ\delta is a topogenous relation, what does AδBA\delta B mean in terms of the corresponding closed subset of βX×βX\beta X \times \beta X?

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 29th 2014

    I believe it means this: given a topogenous relation δ\delta, consider the set

    {(𝒰,𝒱)βX×βX:((A,B)PX×PX)A𝒰B𝒱(A,B)δ}.\{(\mathcal{U}, \mathcal{V}) \in \beta X \times \beta X: (\forall (A, B) \in P X \times P X)\; A \in \mathcal{U} \wedge B \in \mathcal{V} \Rightarrow (A, B) \in \delta\}.

    This is a closed subset of βX×βX\beta X \times \beta X. For example, take δ\delta to be the minimal topogenous relation, consisting of (A,B)(A, B) such that ABA \cap B \neq \emptyset. I claim the corresponding closed subset is the diagonal of βX×βX\beta X \times \beta X. Any point (𝒰,𝒰)(\mathcal{U}, \mathcal{U}) certainly belongs to the subset, since A𝒰A \in \mathcal{U} and B𝒰B \in \mathcal{U} implies AB𝒰A \cap B \in \mathcal{U}, so necessarily ABA \cap B \neq \emptyset. On the other hand, if 𝒰\mathcal{U} and 𝒱\mathcal{V} are distinct ultrafilters, then we can find an A𝒰A \in \mathcal{U} for which A𝒱A \notin \mathcal{V}, so that A𝒰A \in \mathcal{U} and ¬A𝒱\neg A \in \mathcal{V}. Then if (𝒰,𝒱)(\mathcal{U}, \mathcal{V}) belongs to the closed set, we reach the conclusion (A,¬A)δ(A, \neg A) \in \delta or A¬AA \cap \neg A \neq \emptyset, a contradiction.

    (I’ll probably add more to this response in a while; have to run now.)

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 29th 2014
    • (edited Dec 29th 2014)

    Continuing the previous post: let’s check that the subset K δK_\delta of βX×βX\beta X \times \beta X corresponding to a topogenous relation δ\delta is closed, as claimed. Suppose (𝒰,𝒱)K δ(\mathcal{U}, \mathcal{V}) \notin K_\delta, so that there exist A,BPXA, B \in P X such that A𝒰A \in \mathcal{U} and B𝒱B \in \mathcal{V} and (A,B)δ(A, B) \notin \delta. Then, letting [A][A] denote the basic clopen {𝒰:A𝒰}\{\mathcal{U}: A \in \mathcal{U}\}, it seems to be trivially true that the open set [A]×[B][A] \times [B] contains (𝒰,𝒱)(\mathcal{U}, \mathcal{V}) and does not intersect K δK_\delta, so the subset is closed.

    So far we have not said anything about the conditions for δ\delta to be a topogenous relation; we do know that K δK_\delta contains the diagonal if δ\delta contains all pairs (A,B)(A, B) that intersect in a nonempty set. So let’s go in the opposite direction: starting from a closed reflexive relation KβX×βXK \subseteq \beta X \times \beta X, produce a corresponding topogenous relation δ K\delta_K. It seems clear what to do: define

    δ K{(A,B)PX×PX:([A]×[B])K}.\delta_K \coloneqq \{(A, B) \in P X \times P X: ([A] \times [B]) \cap K \neq \emptyset\}.

    Let’s see: the map PXPβXP X \to P\beta X mapping AA to [A][A] is a Boolean algebra map and therefore preserves joins, and the operations ×[B]- \times [B] and K- \cap K also preserve joins, and one pretty quickly concludes from there that the conditions for δ K\delta_K to be a topogenous relation are met.

    We should go on and check that δK δ\delta \mapsto K_\delta is inverse to Kδ KK \mapsto \delta_K, giving a covariant bijection between topogenous relations and reflexive relations on βX\beta X. I think I’ll skip this for now (I’m reasonably confident).

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeDec 30th 2014

    Ok, this is starting to make sense. If we think of points of βX\beta X as “ideal points” of XX, then K δK_\delta is the “ideal set” of all points (x,y)(x,y) such that every pair of sets AxA\ni x and ByB\ni y are δ\delta-related. Except that we have to allow x,yx,y to be in βX\beta X rather than XX for this to retain enough information.

    That makes me wonder what topogenous relations look like in nonstandard analysis. The obvious thing would be to consider the analogous subset of *X×*X{^*X} \times {^*X}, the pairs (x,y)(x,y) such that if x*Ax\in {^*A} and y*By\in {^*B} then AδBA\delta B. This gives a reflexive (external) relation δ\approx_\delta on nonstandard points of XX. Does this relation retain all the information about δ\delta? Can one characterize the external relations which arise from topogenous relations?

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 30th 2014
    • (edited Dec 30th 2014)

    So okay, I might as well go ahead and establish the bijection mentioned in my last message. It will simplify things a bit if we drop the reflexivity from both sides, and just concentrate on conditions 2 and 3 for a topogenous relation (I call such a thing a topogeny), and similarly on the β\beta side drop the reflexivity. It will also work more smoothly if I set up a Galois connection between complements of topogenies (call them antitopogenies) and closed sets of βX×βX\beta X \times \beta X. I’ll remark that if δ\delta is an antitopogeny, then (A,B)δ(A, B) \in \delta and (A,B)δ(A', B) \in \delta together implies (AA,B)δ(A \cup A', B) \in \delta, and similarly in the second argument; also sets (,B)(\emptyset, B) and (A,)(A, \emptyset) belong to δ\delta.

    So, we set up a relation RPX×PX×βX×βXR \subseteq P X \times P X \times \beta X \times \beta X where (A,B,𝒰,𝒱)R(A, B, \mathcal{U}, \mathcal{V}) \in R if ¬A𝒰\neg A \in \mathcal{U} or ¬B𝒱\neg B \in \mathcal{V}. This induces a Galois connection in the usual way, where given a subset δPX×PX\delta \subseteq P X \times P X we consider

    δR{(𝒰,𝒱):((A,B)δ)R(A,B,𝒰,𝒱)}\delta \multimap R \coloneqq \{(\mathcal{U}, \mathcal{V}): (\forall (A, B) \in \delta)\; R(A, B, \mathcal{U}, \mathcal{V})\}

    which is an intersection of basic closed sets ¬([A]×[B])\neg ([A] \times [B]) with (A,B)(A, B) ranging over δ\delta, hence closed. And given a subset KβX×βXK \subseteq \beta X \times \beta X we consider

    KR{(A,B):((𝒰,𝒱)K)R(A,B,𝒰,𝒱)}K \multimap R \coloneqq \{(A, B): (\forall (\mathcal{U}, \mathcal{V}) \in K)\; R(A, B, \mathcal{U}, \mathcal{V})\}

    which is an intersection of antitopogenous relations ¬(𝒰×𝒱)\neg (\mathcal{U} \times \mathcal{V}), hence antitopogenous. With the Galois connection infrastructure in place, it’s enough to show that (1) every closed set of βX×βX\beta X \times \beta X is of the form K δK_\delta for some δPX×PX\delta \subseteq P X \times P X, and (2) every antitopogeny is of the form δ K\delta_K for some KβX×βXK \subseteq \beta X \times \beta X. (1) is trivial since every closed KK is an intersection of basic closed sets of the form ¬([A]×[B])\neg ([A] \times [B]) (i.e., the basic open sets are [A]×[B][A] \times [B]), so just take those (A,B)(A, B) we need to form the set δ\delta. (2) is an application of the ultrafilter lemma. I won’t go through all the details, but if δPX×PX\delta \subseteq P X \times P X is an antitopogeny, then the collection of sets (A,0)δ(A, 0) \in \delta is an ideal and is contained in a maximal ideal, and furthermore one has sufficiently many such maximal ideals (i.e., every ideal is the intersection of the maximal ideals ¬𝒰\neg \mathcal{U} containing it). Similarly sets of the form (0,B)(0, B) form an ideal which is the intersection of sufficiently many maximal ideals, and one can work out that any antitopogeny is an intersection of sets of the form {(A,B):A𝒰orB𝒱}\{(A, B): A \notin \mathcal{U} \;\text{or}\; B \notin \mathcal{V}\} where 𝒰,𝒱\mathcal{U}, \mathcal{V} range over sufficiently many ultrafilters.

    In fact, I’m going to add the remark now to syntopogenous space that every topogeny is a union of “basic topogenies” which are those of the form 𝒰×𝒱PX×PX\mathcal{U} \times \mathcal{V} \subset P X \times P X, a cartesian product of ultrafilters.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeDec 17th 2016

    At syntopogenous space I wrote as the definition:

    • For any δ𝒪\delta\in \mathcal{O}, there exists a δ𝒪\delta'\in\mathcal{O} such that if A,BXA,B\subseteq X have the property that whenever CD=XC\cup D = X, either AδCA\;\delta'\; C or BδDB\;\delta'\; D, then AδBA\;\delta\; B.

    But now that seems wrong to me: instead of BδDB\;\delta'\; D it should be DδBD\;\delta'\; B. That’s what we have in the corresponding condition at proximity space, and that’s the only way I can see to make (for instance) the identification of topological spaces with simple perfect syntopogenies true. So I’ve fixed it. I no longer have Császár’s book in front of me, but I suspect I translated incorrectly from his use of \ll instead of δ\delta. (Actually, it would be nice to rewrite the page treating δ,,\delta,\bowtie,\ll on an equal footing the way we do at proximity space.)

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeDec 29th 2016

    I added some more remarks to syntopogenous space about \bowtie and \ll and how things behave constructively.

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 6th 2017

    Coming back and re-reading Todd’s page on topogenies, it’s clear from the positioning of negations that constructively, the definition using \ll is the most natural. This was also the way Császár originally phrased it; I now regret that I originally wrote syntopogenous space in terms of δ\delta relations instead. Specifically, what Császár calls “topogenous orders” \ll are what Todd calls “flat profunctors”, whose composites are simply relational composites. Moreover, a simple syntopogeny (i.e. a quasi-proximity) is just a comonoid in the bicategory of such profunctors, and an arbitrary syntopogeny should be obtainable similarly from its local ind-completion.

    Why a comonoid? It seems related to the observation that apartness relations are the open complements, in a discrete locale, of closed equivalence relations, so that their “comparison” axiom is more like a comultiplication than a multiplication. So I am wondering whether the characterizations of topogenies and relational beta-modules in terms of the ultrafilter space β\beta could constructively be rephrased using the localic Stone-Cech compactification, whose frame of opens is directly related to ideals and filters without needing to pass through the ultrafilters (its points, of which constructively there may not be enough). Perhaps this would yield the corresponding “apartness” notions.

    However, I’m currently a bit stuck on the constructive counterpart of Todd’s Proposition 4.10 with its slick use of Stone duality. I think if we remove the dualizations, then the result ought to be that opens in βX 1××βX n\beta X_1\times\dots\times \beta X_n correspond to separately join-preserving maps PX 1××PX nΩ opP X_1 \times \cdots \times P X_n \to \Omega^{op} (or perhaps some subclass of these satisfying a “decomposability” property dual to complete regularity). The latter in the case n=2n=2 would be the \bowtie version of a topogenous relation, so that its complement “δ\delta” would be a closed sublocale. Separately join-preserving maps PX 1××PX nΩ opP X_1 \times \cdots \times P X_n \to \Omega^{op} should still correspond to ideals in the distributive lattice PX 1PX nP X_1 \otimes \cdots \otimes P X_n, but where do we go from there?

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 6th 2017

    Hmm, now I am getting a glimmer. There are actually two, or maybe three, different compactifications hiding under the name β\beta. Namely, for any locale XX, we can construct:

    • The locale K 0(X)K_0(X) with O(K 0(X))=Idl(X)O(K_0(X)) = Idl(X), the frame of ideals in XX (which is the free frame on the underlying distributive lattice of XX)
    • The locale K 2(X)K_2(X) with O(K 2(X))O(K_2(X)) the frame of “completely regular” ideals (those for which any UIU\in I is \ll some VIV\in I, where \ll is the “very inside” relation of complete regularity)
    • The locale K 1(X)K_1(X), which is like K 2(X)K_2(X) but using “strongly regular” ideals instead, for which \ll is the finest interpolating relation contained in the “well-inside” relation of regularity

    Countable choice gives K 1(X)=K 2(X)K_1(X) = K_2(X). Moreover, if O(X)O(X) is a Boolean algebra (such as if excluded middle holds and XX is discrete), then K 0(X)=K 1(X)=K 2(X)K_0(X) = K_1(X) = K_2(X) too, since in a Boolean algebra every element is very-inside itself. In general, K 1(X)K_1(X) or K 2(X)K_2(X) is the “Stone-Cech compactification” β\beta; but I suspect that when applying β\beta to discrete spaces/locales as we do when defining relational β\beta-modules and so on, constructively we may be more interested in K 0(X)K_0(X).

    In particular, while K 1K_1 and K 2K_2 are reflections into the full subcategories of compact regular and compact completely-regular locales, K 0K_0 is a non-idempotent monad on LocLoc; it is dual to the comonad on FrmFrm induced by the free-forgetful adjunction to DLatDLat. So it seems more likely to give an interesting notion of generalized multicategory.

    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 9th 2017

    To start with, here is a constructive proof of Todd’s theorem that topogenies from XX to YY correspond to closed subspaces of βX×βY\beta X \times \beta Y. Constructively, I will dualize both the topogenies and the subspaces, and prove that topogenous apartnesses from XX to YY (relations \bowtie between subsets of XX and YY such that ABC(ACBC)A\cup B \bowtie C \iff (A\bowtie C \wedge B\bowtie C) and C\emptyset \bowtie C and dually) correspond to open sublocales of K 0X×K 0YK_0 X \times K_0 Y, where O(K 0(X))=Idl(P(X))O(K_0(X)) = Idl(P(X)) as above. (Note that the strongly closed sublocales are exactly the duals of the open ones, though this is not true for spaces constructively.)

    By definition of locale products as suplattice tensor products, the opens in K 0X×K 0YK_0 X \times K_0 Y are formal joins of tensor products of an ideal in P(X)P(X) with an ideal in P(Y)P(Y), modulo an equivalence relation making the tensor product distribute over joins in both variables. In one direction, given a topogenous apartness \bowtie, we send it to the join of tensor products of principal ideals AB(A)(B)\bigvee_{A\bowtie B} \downarrow(A) \otimes \downarrow(B). In the other direction, given an open i(J iK i)\bigvee_i (J_i\otimes K_i), we define ABA\bowtie B to mean A=A 1A nA = A_1 \cup \cdots \cup A_n and B=B 1B mB = B_1 \cup \cdots \cup B_m where for each j,kj,k there is an ii such that A jJ iA_j \in J_i and B kK iB_k \in K_i.

    To show that these are inverses, starting from \bowtie, if A=A 1A nA = A_1 \cup \cdots \cup A_n and B=B 1B mB = B_1 \cup \cdots \cup B_m and for each j,kj,k there exist principal ideals (C)\downarrow(C) and (D)\downarrow(D) with A jCA_j \subseteq C and B kDB_k \subseteq D and CDC\bowtie D, then also A jB kA_j \bowtie B_k by isotony, hence ABA\bowtie B by finite additivity of \bowtie. (And obviously, if ABA\bowtie B then A(A)A\in \downarrow(A) and B(B)B\in \downarrow (B).)

    On the other side, since any ideal is the directed join of the principal ideals it contains (directed joins of ideals are just unions) and \otimes distributes over joins, any open is equivalent to a join of tensor products i(C i)(D i)\bigvee_i \downarrow(C_i) \otimes \downarrow (D_i) of tensor products of principal ideals. Then each C iD iC_i \bowtie D_i in the resulting apartness \bowtie, hence (C i)(D i)\downarrow(C_i) \otimes \downarrow (D_i) appears in the open reconstructed from it. Conversely, if ABA\bowtie B, then A=A 1A nA = A_1 \cup \cdots \cup A_n and B=B 1B mB = B_1 \cup \cdots \cup B_m and for each j,kj,k there exists an i j,ki_{j,k} with A jC i j,kA_j \subseteq C_{i_{j,k}} and B kD i j,kB_k \subseteq D_{i_{j,k}}. Then by definition of the finite join of ideals, the ideal J= j,k(C i j,k)J = \bigvee_{j,k} \downarrow(C_{i_{j,k}}) contains AA and the ideal K= j,k(D i j,k)K = \bigvee_{j,k} \downarrow(D_{i_{j,k}}) contains BB, and the tensor product JKJ\otimes K might as well appear in the given open since \otimes distributes over joins, hence so might as well also (A)(B)\downarrow(A)\otimes \downarrow(B).

    A similar, but simpler, argument shows that opens in X×K 0(Y)X\times K_0(Y) correspond to point-set apartness relations xAx\bowtie A for xXx\in X and AYA\subseteq Y satisfying isotony and finite additivity on the right.

    • CommentRowNumber22.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 9th 2017

    I haven’t read through this properly yet, but this sounds awesome!

    I never tried to consider the constructive aspects while writing the topogeny stuff up, because the stuff I was trying to reinterpret seemed so inundated with excluded middle in the first place (and I felt sure the guy I was communicating with at the time would not be the least interested). So I’m glad you’re taking this up.

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 10th 2017

    Perhaps it’s better to consider the opens in K 0X×K 0YK_0 X \times K_0 Y as (-1)-sheaves for a coverage, the way Johnstone does it. Then they are sets 𝒥\mathcal{J} of tensor products JKJ\otimes K, where JIdl(P(X))J\in Idl(P(X)) and KIdl(P(Y))K\in Idl(P(Y)), such that

    1. if J 2K 2𝒥J_2\otimes K_2\in \mathcal{J} and J 1J 2J_1 \subseteq J_2 and K 1K 2K_1\subseteq K_2, then J 1K 1𝒥J_1\otimes K_1 \in \mathcal{J} (presheaf)
    2. if J iK𝒥J_i \otimes K \in \mathcal{J} for all ii then ( iJ i)K𝒥(\bigvee_i J_i)\otimes K \in \mathcal{J} and dually, where iJ i={A 1A n\bigvee_i J_i = \{ A_1 \vee \cdots \vee A_n \mid each A iA_i belongs to some J i}J_i \} (sheaf)

    With this representation, the apartness relation corresponding to an open 𝒥\mathcal{J} says that ABA\bowtie B if (A)(B)𝒥\downarrow(A) \otimes \downarrow(B) \in \mathcal{J}, and the open corresponding to an apartness relation is {JKAJ,BK,AB}\{ J\otimes K \mid \forall A\in J, \forall B\in K, A\bowtie B\}. The bijection between the two is then even easier.

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 10th 2017

    Now I claim that just like apartness relations are exactly closed equivalence relations on a discrete locale, decomposable proximal (set-set) apartnesses are exactly closed equivalence relations on K 0XK_0 X for a discrete locale XX (and similarly decomposable quasi-proximal apartnesses correspond to closed preorders). The above argument shows that closed sublocales of K 0X×K 0XK_0 X \times K_0 X correspond to set-set relations \bowtie satisfying the appropriate isotony and additivity properties, and the correspondence of symmetry is clear; so it suffices to consider reflexivity and decomposable transitivity.

    For reflexivity, the diagonal is contained in a closed sublocale CU\mathsf{C}U of K 0X×K 0XK_0 X \times K_0 X iff UU is contained in the exterior of the diagonal, which is the largest open whose restriction along the diagonal is 00. As a (-1)-sheaf, this exterior is the set of all JKJ\otimes K such that JK=0J\cap K = 0. Thus, CU\mathsf{C}U contains the diagonal iff ABA\bowtie B implies AB=0A\cap B = 0, which is the proximal reflexivity axiom.

    Transitivity of CU\mathsf{C}U means that the pullback π 2 *U\pi_2^* U is contained in π 1 *Uπ 3 *U\pi_1^* U \vee \pi_3^* U as open subsets of K 0X×K 0X×K 0XK_0 X \times K_0 X \times K_0 X. If UU corresponds to \bowtie, then we have

    π 1 *U={JKLBK,CL,BC} \pi_1^* U = \{ J\otimes K\otimes L \mid \forall B\in K, \forall C\in L, B\bowtie C \} π 2 *U={JKLAJ,CL,AC} \pi_2^* U = \{ J\otimes K\otimes L \mid \forall A\in J, \forall C\in L, A\bowtie C \} π 3 *U={JKLAJ,BK,AB} \pi_3^* U = \{ J\otimes K\otimes L \mid \forall A\in J, \forall B\in K, A\bowtie B \}

    The tricky thing is π 1 *Uπ 3 *U\pi_1^* U \vee \pi_3^* U, since it is not just the union of these sets but the (-1)-sheafification of the union. However, I claim that

    π 1 *Uπ 3 *U={JKLAJ,BK,CL,D,E,(ADECB=DE}\pi_1^* U \vee \pi_3^* U = \{ J\otimes K\otimes L \mid \forall A\in J, \forall B\in K, \forall C\in L, \exists D,E, (A\bowtie D \wedge E\bowtie C \wedge B = D\cup E \}

    It’s easy to see that this is a (-1)-sheaf, and that it contains π 1 *U\pi_1^*U and π 3 *U\pi_3^*U. Moreover, every JKLJ\otimes K\otimes L in this set can be obtained from those in π 1 *U\pi_1^*U and π 3 *U\pi_3^*U by three steps of gluing:

    JKL = AJ((A)KL) = AJ CL((A)K(C)) = AJ CL((A)K A(C)) AJ CL((A)K C(C))\begin{aligned} J\otimes K \otimes L &= \bigvee_{A\in J} (\downarrow(A) \otimes K\otimes L)\\ &= \bigvee_{A\in J} \bigvee_{C\in L} (\downarrow(A) \otimes K \otimes \downarrow(C))\\ &= \bigvee_{A\in J} \bigvee_{C\in L} (\downarrow(A) \otimes K_A \otimes \downarrow(C)) \vee \bigvee_{A\in J} \bigvee_{C\in L} (\downarrow(A) \otimes K_C \otimes \downarrow(C)) \end{aligned}

    where K A={DKAD}K_A = \{ D\in K \mid A\bowtie D\} and K C={EKEC}K_C = \{ E \in K \mid E\bowtie C \}, so that (A)K A(C)π 3 *U\downarrow(A) \otimes K_A \otimes \downarrow(C) \in \pi_3^*U and (A)K C(C)π 1 *U\downarrow(A) \otimes K_C \otimes \downarrow(C)\in \pi_1^*U. The final decomposition as a binary join uses the assumed property of JKLJ\otimes K\otimes L and the fact that joins of ideals are computed by taking finite joins of their elements. Thus, the above set of triples of ideals is a (-1)-sheaf containing π 1 *U\pi_1^*U and π 3 *U\pi_3^*U and contained in the sheafification of their union, so it must be their join as (-1)-sheaves.

    Now transitivity π 2 *Uπ 1 *Uπ 3 *U\pi_2^* U\subseteq \pi_1^* U \vee \pi_3^* U is equivalent to the special case that each triple of the form (A)(X)(C)\downarrow(A) \otimes \downarrow(X) \otimes \downarrow(C), with ACA\bowtie C, is an element of the above set. This means exactly that whenever ACA\bowtie C there are sets D,ED,E such that ADA\bowtie D and ECE\bowtie C and X=DEX = D\cup E, the decomposability axiom for a proximal apartness.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 10th 2017

    None of this actually requires XX to be discrete. However, if we allow XX to be non-discrete, then there should be some compatibility condition on a proximity, such as that every AO(X)A\in O(X) satisfies A={BO(X)C,AC=XBC}A = \bigvee \{ B\in O(X) \mid \exists C, A\cup C = X \wedge B\bowtie C \}. I’m not sure yet how to interpret that in terms of a closed equivalence relation.

    However, K 0XK_0X has the interesting property that any map f:XYf:X\to Y with YY compact Hausdorff (i.e. compact regular) factors through a map g:K 0XYg:K_0 X \to Y defined by g *(W)={AO(X)Af *W}g^*(W) = \{ A\in O(X) \mid A \triangleleft f^*W \}, where \triangleleft is the well-inside relation of regularity. I don’t think gg is unique, and while K 0XK_0 X itself is compact and contains XX densely, it is not Hausdorff — but quotienting it by a closed equivalence relation seems like a natural way to make it Hausdorff! Certainly the kernel pair of a Hausdorff quotient is a closed equivalence relation, so that if an effective equivalence relation has a Hausdorff quotient then it is closed. Dually, if a closed equivalence relation is effective, then the quotient map is proper, and since closed sublocales descend along proper maps it follows that the quotient is Hausdorff. Thus, the effective closed equivalence relations on K 0XK_0 X correspond bijectively to quotients of K 0XK_0 X that are Hausdorff, giving a locale-theoretic approach to the standard bijective correspondence between proximities and compactifications.

    This suggests strongly that effectiveness should correspond to the above compatibility condition on proximities. However, I don’t know how to show this because I don’t know a characterization of which equivalence relations in LocLoc are effective.

    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 10th 2017

    Oh, that’s silly – the compatibility condition doesn’t mean Hausdorffness of the quotient, it means that the composite map from XX to the quotient is a sublocale embedding. Maybe the quotient is always Hausdorff. I think I can show that it does look like other definitions of the compactification of a proximity locale.

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