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.
   • CommentAuthorMike Shulman
   • CommentTimeNov 22nd 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexSuppose we have a cohesive $(\infty,1)$-topos $\mathbf{H}$. We've established that at least in most examples, the shape modality $ʃ$ can be extended to a stable factorization system, i.e. to an internal modality, which therefore supplies a reflective subcategory of "discrete objects" in each slice $\mathbf{H}/X$. However, we don't know how to extend $\flat$ to an operation of any sort on slices, let alone one that coreflects into this subcategory. Here's a simpler question: if $Y$ is an object of $\mathbf{H}$, we have the coreflection $\flat Y \to Y$, and we can pull it back to $\mathbf{H}/X$. Is the resulting map $X\times \flat Y \to X\times Y$ a coreflection of $X\times Y$ into the discrete objects of $\mathbf{H}/X$?

   Suppose we have a cohesive $(\infty,1)$-topos $\mathbf{H}$. We’ve established that at least in most examples, the shape modality $ʃ$ can be extended to a stable factorization system, i.e. to an internal modality, which therefore supplies a reflective subcategory of “discrete objects” in each slice $\mathbf{H}/X$. However, we don’t know how to extend $\flat$ to an operation of any sort on slices, let alone one that coreflects into this subcategory.

   Here’s a simpler question: if $Y$ is an object of $\mathbf{H}$, we have the coreflection $\flat Y \to Y$, and we can pull it back to $\mathbf{H}/X$. Is the resulting map $X\times \flat Y \to X\times Y$ a coreflection of $X\times Y$ into the discrete objects of $\mathbf{H}/X$?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeNov 23rd 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThe answer is no. In fact, I now have a proof (checked in Coq) that fiberwise coreflections _cannot_ exist, and it only needs coreflections of this sort. More precisely, any "coreflective subuniverse" in HoTT is of the form $(-\times U)$ for some hprop $U$. In particular, if it preserves $1$ (as it would have to, if it coreflected into the discrete objects), then it is the identity.

   The answer is no.

   In fact, I now have a proof (checked in Coq) that fiberwise coreflections cannot exist, and it only needs coreflections of this sort. More precisely, any “coreflective subuniverse” in HoTT is of the form $(-\times U)$ for some hprop $U$. In particular, if it preserves $1$ (as it would have to, if it coreflected into the discrete objects), then it is the identity.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 23rd 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIs there a 'moral' to this story? Perhaps one where we come to see the inevitability of why we can't have what we wanted, but realise that there's a better path to take anyway. Something like the one of overcoming the disappointment of not being able to reach a contradiction from denying the parallel postulate by realising that this shows a richer array of geometries.

   Is there a ’moral’ to this story? Perhaps one where we come to see the inevitability of why we can’t have what we wanted, but realise that there’s a better path to take anyway. Something like the one of overcoming the disappointment of not being able to reach a contradiction from denying the parallel postulate by realising that this shows a richer array of geometries.

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeNov 23rd 2014
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexIs this `categorical pragmatism'?

   Is this ‘categorical pragmatism’?

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 23rd 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThat sounds close to an oxymoron, at least if we're following in the path of Grothendieck (and you mean pragmatism in the popular sense).

   That sounds close to an oxymoron, at least if we’re following in the path of Grothendieck (and you mean pragmatism in the popular sense).

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeNov 23rd 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI wish there were a moral like that. But if there is, I don't think I've seen it yet.

   I wish there were a moral like that. But if there is, I don’t think I’ve seen it yet.