Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMarc
   • CommentTimeOct 24th 2023
   • PermaLink
   Author: Marc
   Format: MarkdownItexOld discussion: +-- {: .query} [[Mike Shulman|Mike]]: Back atcha... do you have any good examples of gauge spaces that are not of one of these types? And is there any value in embedding metric spaces, uniform spaces, and topological spaces into this mysterious larger category $QGau$, in ways so that their images are essentially disjoint? Do we ever, for instance, want to talk about a short map from a metric space to a topological space, or vice versa? I would like the answer to be "yes," but I haven't managed to make it come out that way myself yet. _Toby_: I don\'t know any examples of gauge spaces that arise naturally (you can make them artificially, of course, say as disjoint unions) and don\'t correspond to some other more familiar type of space. However, I can give examples that don\'t belong to $Met$, $Unif$, or $Top$ ... because they\'re Cauchy spaces, which aren\'t listed above yet! Incidentally, I didn\'t list Cauchy spaces yet (and didn\'t finish describing general topological spaces), since I haven\'t checked yet that things behave correctly. (In particular, I haven\'t checked that $Top$ becomes a full subcategory of $QGau$ ---did you?---, although I hope that it will.) I think that it\'s nice to be able to see these all as special cases of one kind of thing; then the usual ways of finding the 'underlying' foo of a bar become reflections (and sometimes coreflections). This is basically how Lowen-Colebunder\'s book (which I\'ve referenced, for example, on [[convergence space]]) works (although her big category of everything is the category of mereological spaces, which includes $Conv$, rather than $QGau$). Seeing this as inclusions and (co)reflections prevents any expectation that the diagrams above should commute, since they mix different things in the wrong way. On the other hand, it also shows us that (for example) any topological space has an underlying uniform space ... if you want it. [[Mike Shulman|Mike]]: I believe I did check that $Top\to QGau$ is full, but you should verify it. I removed my comments about triangles commuting; go ahead and write about the reflections and I'll see whether I still want to make that comment. I can see thinking of uniform spaces as "saturated" gauge spaces, so that the uniform space underlying a metric space is its "saturation" (reflection) in the larger category $Gau$. Perhaps Cauchy spaces can also be thought of this way. I will be kind of surprised if $Top$ turns out to be reflective in $QGau$, but if it were that would also be pretty neat. [[Mike Shulman|Mike]]: I added a new embedding of $Top$ in $QGau$, which seems to me more likely to be (co)reflective. _Toby_: Ah yes, I think that that\'s the good one that I wasn\'t finding. =-- <a href="https://ncatlab.org/nlab/revision/diff/gauge+space/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/gauge+space/15">v15</a>, <a href="https://ncatlab.org/nlab/show/gauge+space">current</a>

   Old discussion:

   +– {: .query} Mike: Back atcha… do you have any good examples of gauge spaces that are not of one of these types? And is there any value in embedding metric spaces, uniform spaces, and topological spaces into this mysterious larger category $QGau$, in ways so that their images are essentially disjoint? Do we ever, for instance, want to talk about a short map from a metric space to a topological space, or vice versa? I would like the answer to be “yes,” but I haven’t managed to make it come out that way myself yet.

   Toby: I don't know any examples of gauge spaces that arise naturally (you can make them artificially, of course, say as disjoint unions) and don't correspond to some other more familiar type of space. However, I can give examples that don't belong to $Met$, $Unif$, or $Top$ … because they're Cauchy spaces, which aren't listed above yet!

   Incidentally, I didn't list Cauchy spaces yet (and didn't finish describing general topological spaces), since I haven't checked yet that things behave correctly. (In particular, I haven't checked that $Top$ becomes a full subcategory of $QGau$ —did you?—, although I hope that it will.)

   I think that it's nice to be able to see these all as special cases of one kind of thing; then the usual ways of finding the ’underlying’ foo of a bar become reflections (and sometimes coreflections). This is basically how Lowen-Colebunder's book (which I've referenced, for example, on convergence space) works (although her big category of everything is the category of mereological spaces, which includes $Conv$, rather than $QGau$).

   Seeing this as inclusions and (co)reflections prevents any expectation that the diagrams above should commute, since they mix different things in the wrong way. On the other hand, it also shows us that (for example) any topological space has an underlying uniform space … if you want it.

   Mike: I believe I did check that $Top\to QGau$ is full, but you should verify it.

   I removed my comments about triangles commuting; go ahead and write about the reflections and I’ll see whether I still want to make that comment. I can see thinking of uniform spaces as “saturated” gauge spaces, so that the uniform space underlying a metric space is its “saturation” (reflection) in the larger category $Gau$. Perhaps Cauchy spaces can also be thought of this way. I will be kind of surprised if $Top$ turns out to be reflective in $QGau$, but if it were that would also be pretty neat.

   Mike: I added a new embedding of $Top$ in $QGau$, which seems to me more likely to be (co)reflective.

   Toby: Ah yes, I think that that's the good one that I wasn't finding. =–

   diff, v15, current