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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 13th 2010
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeOct 13th 2010
    Though the terminology codiscrete may be better and is in some use among category theorists, among other mathematicians the other two terms seem to be much more common (indiscrete and trivial). (I am not complaining, just noting.)
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 13th 2010

    Yes, right. I did mention these terms, though. But I think here we may play Bourbaki a bit. I am really paving the way for a comprehensive discussion of discrete and codiscrete spaces at cohesive topos.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeOct 13th 2010
    I am sure you are going to describe a magnificent picture...just go on nondistracted :)

    I am giving a talk in Vienna on a different topic (key words: Lie algebroids, Leibniz algebras, symmetrization maps, realizations and deformation quantization). I am a bit at the edge of getting sick before the travel. Need some rest before...
    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeOct 14th 2010

    I observe that discrete topology redirects to discrete space, but codiscrete topology redirects to discrete and codiscrete topology. Probably either the first should redirect to discrete and codiscrete topology or the second should redirect to codiscrete space?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 14th 2010

    Fixed. i made everything referring specifically to topological spaces redirect to discrete and codiscrete topology.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2019

    added mentioning of the alternative name “chaotic topology” for “indiscrete topology”, and pointer to places that use it (so far: Stacks Projcect 7.6.6, but eventually there should be more canonical pointers)

    Incidentally, the Wikipedia entry on Grothendieck topologies currently mixes up the terminology here. Somebody should fix this

    diff, v7, current

    • CommentRowNumber8.
    • CommentAuthorAli Caglayan
    • CommentTimeMay 26th 2019

    I seem to recall “chaotic preorder” stands for the right adjoint of the forgetful functor from PreorderSetPreorder \to Set, and perhaps elsewhere too. It seems a bit silly to call the indiscrete topology chaotic however, even though it is a right adjoint.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 26th 2019

    I’m not sure who introduced the term; might it have been Lawvere? Here is one source of discussion: Lawvere.

    In a footnote here, page 3, completely random motion (chaos) is opposed to immobility (discreteness), where open sets cannot distinguish between point-particles in motion in the chaotic case.

    Google searches confirm that the terminology continues to be used to this day, so the nLab should keep the terminology on record.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 27th 2019
    • (edited May 27th 2019)

    added the pointer to Lawvere’s “Functorial remarks on the general concept of chaos”, and am also adding it at codiscrete space and at chaos

    diff, v9, current

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 23rd 2020

    Corrected the footnote reference in Lawvere 86.

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 23rd 2020

    Added the SGA4 origin of chaotic topology.

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 28th 2021
    • (edited Jan 28th 2021)

    Does the functor

    Disc: Set→Top

    that equips a set with its discrete topology admit a left adjoint functor?

    Initially I thought that the answer should be trivially no, but Disc appears to preserve all small limits and the solution set condition seems to be satisfied because any topological space has only a set of isomorphism classes of surjective continuous maps to discrete spaces.

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 28th 2021

    Never mind, of course Disc does not preserve infinite products.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJan 28th 2021

    Just thinking that the would-be left adjoint would have to form sets of “quasi-components”, as remarked here. So what goes wrong? Is assigning quasi-components not functorial?

    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 28th 2021

    Re #15: the relation “x∼ qyx \sim_q y iff f(x)=f(y)f(x) = f(y) for every continuous map f:X→Df \colon X \to D to a discrete space DD.” does appear give a functor Top→Set, since if x~y, then also g(x)~g(y) for any continuous map g, since for any continuous map h to a discrete space, hg is also a continuous map to a discrete space.

    But it does not appear to satisfy the adjunction relation for pathological spaces.

    Also, I doubt that the quasi-components functor preserves coequalizers.

    • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 28th 2021

    If you replace spaces by locally connected spaces, then the discrete space functor does have a left adjoint.

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 28th 2021

    Recorded the content of the above discussion in a new subsection.

    diff, v13, current

    • CommentRowNumber19.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 28th 2021

    “Indiscrete topology” has 34 hits on MathSciNet, “antidiscrete topology” has 1 hits, “codiscrete topology” has 0 hits.

    I renamed the article to reflect the dominant terminology, but added mentions of other names.

    diff, v13, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJan 29th 2021

    I have added the pointer to where the cohesion of locally connected spaces is discussed (here).

    But it seemed you were after something more general. Is locally connected spaces the largest class for which the left adjoint to Disc exists? Why does assigning quasi-components not work more generally?

    diff, v14, current

    • CommentRowNumber21.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 18th 2021

    Re #20: This is a good question, which I’d like to think about when I have a quiet moment to myself.

    • CommentRowNumber22.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 18th 2021
    • (edited Feb 18th 2021)

    Re #20 and #21: I think that Johnstone’s result mentioned at locally connected topos, applied to the big topos of (some chosen full subcategory of) topological spaces, probably shows that the existence of the left adjoint characterises exactly locally connected spaces, i.e. the latter are indeed the largest possible class for which the left adjoint exists.

    • CommentRowNumber23.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 18th 2021
    • (edited Feb 18th 2021)

    I think that what goes wrong for quasi-components is exactly what is described in Warning 2.7 at connected component. The category-theoretic notion uses coproducts fundamentally.

    • CommentRowNumber24.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 18th 2021

    Let’s see, I think there are several questions at play. We are contemplating full subcategories i:𝒮Topi: \mathcal{S} \hookrightarrow Top through which the full inclusion Δ:SetTop\Delta: Set \to Top (the discrete space functor) factors, so that each functor 𝒮(X,Δ)\mathcal{S}(X, \Delta -) is representable. This functor is Top(X,Δ)Top(X, \Delta-), by full faithfulness of ii. So the full subcategory of TopTop consisting of all spaces XX such that Top(X,Δ):SetSetTop(X, \Delta-): Set \to Set is representable is the largest class/subcategory 𝒮\mathcal{S} that Urs is speaking of.

    Spaces XX that are coproducts of connected spaces, i.e., topological coproducts of their connected components, may be just what we are looking for. These are not the same as locally connected spaces! For example, a connected space need not be locally connected, the topologists’ sine curve being a classic example.

    Certainly if XX is connected, then Top(X,Δ):SetSetTop(X, \Delta-): Set \to Set is isomorphic to the identity functor IdId, because (letting SAS \cdot A denote an SS-indexed coproduct of copies of AA) we have natural isomorphisms

    Top(X,ΔS)Top(X,SΔ1)STop(X,Δ1)S1STop(X, \Delta S) \cong Top(X, S \cdot \Delta 1) \cong S \cdot Top(X, \Delta 1) \cong S \cdot 1 \cong S

    where the second isomorphism uses the definition of connected space, that Top(X,)Top(X, -) preserves coproducts. And then we would have, for a family X iX_i of connected spaces, that

    Top( i:TX i,Δ) i:TTop(X i,Δ)Id T=hom(T,)Top(\sum_{i: T} X_i, \Delta -) \cong \prod_{i:T} Top(X_i, \Delta -) \cong Id^T = \hom(T, -)

    so that i:TX i\sum_{i: T} X_i falls within our class.

    Now I’d like to sketch an argument that this is the precise class of Urs’s spaces XX. Here we have an adjoint string of functors between 𝒮\mathcal{S} and SetSet, which I’ll denote as

    ΠΔΓ\Pi \dashv \Delta \dashv \Gamma \dashv \nabla

    without assuming a priori that Π\Pi is the connected components functor, but I intend to prove that it is. First, the unit as continuous map u:XΔΠXu: X \to \Delta \Pi X is surjective: this is equivalent to saying that the canonical map ΓXΠX\Gamma X \to \Pi X is epic, but this is equivalent to the canonical map Δ\Delta \to \nabla (from discrete to indiscrete) being monic. Each point pΔΠXp \in \Delta \Pi X is clopen, so the inverse images u 1(p)u^{-1}(p) are disjoint clopen subsets of XX, and so we have a topological coproduct decomposition

    X p:ΠXu 1(p)X \cong \sum_{p: \Pi X} u^{-1}(p)

    and the projections

    Top(X,Δ)Top( p:ΠXu 1(p),Δ) p:ΠXTop(u 1(p),Δ)Top(u 1(p),Δ)Top(X, \Delta-) \cong Top(\sum_{p: \Pi X} u^{-1}(p), \Delta -) \cong \prod_{p: \Pi X} Top(u^{-1}(p), \Delta-) \to Top(u^{-1}(p), \Delta -)

    must match the evident projection maps

    Set(ΠX,)Set(p:1ΠX,)Set(1,)IdSet(\Pi X, -) \stackrel{Set(p: 1 \to \Pi X, -)}{\longrightarrow} Set(1, -) \cong Id

    i.e., we must have Top(u 1(p),Δ)IdTop(u^{-1}(p), \Delta -) \cong Id. I will show this forces each u 1(p)u^{-1}(p) to be connected.

    It turns out this is really easy. Let 22 denote a 2-element set. For any nonempty space AA, there is an obvious injection 2Top(A,Δ2)2 \hookrightarrow Top(A, \Delta 2). If u 1(p)u^{-1}(p) were disconnected, say as a topological coproduct u 1(p)=A+Bu^{-1}(p) = A + B, we would have

    2×2Top(A,Δ2)×Top(B,Δ2)Top(A+B,Δ2)Top(u 1(p),Δ2)22 \times 2 \hookrightarrow Top(A, \Delta 2) \times Top(B, \Delta 2) \cong Top(A + B, \Delta 2) \cong Top(u^{-1}(p), \Delta 2) \cong 2

    and this is a clear contradiction.

    • CommentRowNumber25.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 18th 2021

    As far as quasi-components go: if XX is a coproduct x:ΠXC(x)\sum_{x: \Pi X} C(x) where C(x)C(x) is the connected component of some representative point xx, then C(x)QC(x)C(x) \subseteq QC(x) where QC(x)QC(x) is the quasi-component of xx. This quasi-component is the intersection of all clopens containing xx. But C(x)C(x) is already clopen in the coproduct decomposition, so QC(x)C(x)QC(x) \subseteq C(x). Hence for the class of spaces Urs was asking about, there is no difference between taking connected components and quasi-components.

    There was a question in #15 about functoriality of Q:TopSetQ: Top \to Set, taking a space XX to its set of quasi-components. All this might be well-known, but I didn’t know, so let’s see: given a continuous map f:XYf: X \to Y, suppose f(x)=yf(x) = y. We want to show that f(QC(x))QC(y)f(QC(x)) \subseteq QC(y). This is the same as QC(x)f 1(QC(y))QC(x) \subseteq f^{-1}(QC(y)). The inverse image of each clopen containing yy is a clopen containing xx, and the inverse image of the intersection of all clopens containing yy will be an intersection of clopens containing xx, and QC(x)QC(x) will be contained in this intersection. So functoriality goes through.

    • CommentRowNumber26.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 18th 2021
    • (edited Feb 19th 2021)

    Spaces X that are coproducts of connected spaces, i.e., topological coproducts of their connected components, may be just what we are looking for. These are not the same as locally connected spaces!

    Hehe, indeed! Yes, I was thinking of coproducts of connected spaces as the characterising class. I was kind of applying Warning 2.7 that I mentioned the wrong way around in this regard!

    I haven’t looked closely at your proof yet, but I wouldn’t be surprised if it unravels to Johnstone’s proof (or conversely!).

    • CommentRowNumber27.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 18th 2021

    Simple question: are these spaces where Π\Pi makes sense precisely the ones where quasi-components and components coincide?

    • CommentRowNumber28.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 18th 2021

    David: I don’t believe so. Compact Hausdorff spaces are another class where they coincide.

    • CommentRowNumber29.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 19th 2021

    OK, cool, thanks.

    • CommentRowNumber30.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 12th 2021
    Re #20: One natural answer to this is just like the adjunction
    between presheaves on X and locales over X restricts
    to an equivalence between sheaves on X and etale maps to X,
    the adjunction between locally connected locales over X and precosheaves over X
    restricts to an equivalence between complete spreads over X and cosheaves over X.
    This is documented at cosheaf of connected components and display locale.
    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeAug 25th 2021

    Re-organized the references to make them appear in chronological order.

    diff, v16, current

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