Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
created discrete and codiscrete topology
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.
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?
Fixed. i made everything referring specifically to topological spaces redirect to discrete and codiscrete topology.
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
I seem to recall “chaotic preorder” stands for the right adjoint of the forgetful functor from $Preorder \to Set$, and perhaps elsewhere too. It seems a bit silly to call the indiscrete topology chaotic however, even though it is a right adjoint.
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.
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
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.
Never mind, of course Disc does not preserve infinite products.
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?
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.
If you replace spaces by locally connected spaces, then the discrete space functor does have a left adjoint.
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?
Re #20: This is a good question, which I’d like to think about when I have a quiet moment to myself.
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.
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.
Let’s see, I think there are several questions at play. We are contemplating full subcategories $i: \mathcal{S} \hookrightarrow Top$ through which the full inclusion $\Delta: Set \to Top$ (the discrete space functor) factors, so that each functor $\mathcal{S}(X, \Delta -)$ is representable. This functor is $Top(X, \Delta-)$, by full faithfulness of $i$. So the full subcategory of $Top$ consisting of all spaces $X$ such that $Top(X, \Delta-): Set \to Set$ is representable is the largest class/subcategory $\mathcal{S}$ that Urs is speaking of.
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! For example, a connected space need not be locally connected, the topologists’ sine curve being a classic example.
Certainly if $X$ is connected, then $Top(X, \Delta-): Set \to Set$ is isomorphic to the identity functor $Id$, because (letting $S \cdot A$ denote an $S$-indexed coproduct of copies of $A$) we have natural isomorphisms
$Top(X, \Delta S) \cong Top(X, S \cdot \Delta 1) \cong S \cdot Top(X, \Delta 1) \cong S \cdot 1 \cong S$where the second isomorphism uses the definition of connected space, that $Top(X, -)$ preserves coproducts. And then we would have, for a family $X_i$ of connected spaces, that
$Top(\sum_{i: T} X_i, \Delta -) \cong \prod_{i:T} Top(X_i, \Delta -) \cong Id^T = \hom(T, -)$so that $\sum_{i: T} X_i$ falls within our class.
Now I’d like to sketch an argument that this is the precise class of Urs’s spaces $X$. Here we have an adjoint string of functors between $\mathcal{S}$ and $Set$, which I’ll denote as
$\Pi \dashv \Delta \dashv \Gamma \dashv \nabla$without assuming a priori that $\Pi$ is the connected components functor, but I intend to prove that it is. First, the unit as continuous map $u: X \to \Delta \Pi X$ is surjective: this is equivalent to saying that the canonical map $\Gamma X \to \Pi X$ is epic, but this is equivalent to the canonical map $\Delta \to \nabla$ (from discrete to indiscrete) being monic. Each point $p \in \Delta \Pi X$ is clopen, so the inverse images $u^{-1}(p)$ are disjoint clopen subsets of $X$, and so we have a topological coproduct decomposition
$X \cong \sum_{p: \Pi X} u^{-1}(p)$and the projections
$Top(X, \Delta-) \cong Top(\sum_{p: \Pi X} u^{-1}(p), \Delta -) \cong \prod_{p: \Pi X} Top(u^{-1}(p), \Delta-) \to Top(u^{-1}(p), \Delta -)$must match the evident projection maps
$Set(\Pi X, -) \stackrel{Set(p: 1 \to \Pi X, -)}{\longrightarrow} Set(1, -) \cong Id$i.e., we must have $Top(u^{-1}(p), \Delta -) \cong Id$. I will show this forces each $u^{-1}(p)$ to be connected.
It turns out this is really easy. Let $2$ denote a 2-element set. For any nonempty space $A$, there is an obvious injection $2 \hookrightarrow Top(A, \Delta 2)$. If $u^{-1}(p)$ were disconnected, say as a topological coproduct $u^{-1}(p) = A + B$, we would have
$2 \times 2 \hookrightarrow Top(A, \Delta 2) \times Top(B, \Delta 2) \cong Top(A + B, \Delta 2) \cong Top(u^{-1}(p), \Delta 2) \cong 2$and this is a clear contradiction.
As far as quasi-components go: if $X$ is a coproduct $\sum_{x: \Pi X} C(x)$ where $C(x)$ is the connected component of some representative point $x$, then $C(x) \subseteq QC(x)$ where $QC(x)$ is the quasi-component of $x$. This quasi-component is the intersection of all clopens containing $x$. But $C(x)$ is already clopen in the coproduct decomposition, so $QC(x) \subseteq C(x)$. 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 $Q: Top \to Set$, taking a space $X$ 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 $f: X \to Y$, suppose $f(x) = y$. We want to show that $f(QC(x)) \subseteq QC(y)$. This is the same as $QC(x) \subseteq f^{-1}(QC(y))$. The inverse image of each clopen containing $y$ is a clopen containing $x$, and the inverse image of the intersection of all clopens containing $y$ will be an intersection of clopens containing $x$, and $QC(x)$ will be contained in this intersection. So functoriality goes through.
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!).
Simple question: are these spaces where $\Pi$ makes sense precisely the ones where quasi-components and components coincide?
David: I don’t believe so. Compact Hausdorff spaces are another class where they coincide.
OK, cool, thanks.
1 to 31 of 31