Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 finite 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 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: condensed infinity-groupoid

Bottom of Page

1 to 7 of 7

  1.  

  1. hopefully the given definition makes sense and is equivalent to the definition found in Scholze’s “Lectures on analytic geometry”, somebody more knowledgeable at (∞,1)(\infty,1)-category theory could double check.

    Anonymous

    v1, current

  2. adding note about terminology

    Anonymous

    v1, current

  3. According to the Algebraic Topology discord server, both definitions have size issues (one only has condensed infinity-groupoids that are small relative to a universe/a strong limit cardinal). Other than that, the first definition (“a (infinity,1)-sheaf of infinity-groupoids on the pro-étale (infinity,1)-site of the point”) is correct, but the other definition is incorrect: the sheaf condition is wrong, it should be profinite spaces (pro-objects in finite infinity-groupoids/finite sets) rather than pro-infinity-groupoids, and a hyperdescent condition is also required.

  4. Peter Scholze said that condensed infinity-groupoids seem to form an elementary (∞,1)(\infty,1)-topos

    Anonymous

    diff, v6, current

  5. adding ideas section

    Anonymous

    diff, v6, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2022
    • (edited Jul 5th 2022)

    The lead-in paragraph does not make good sense to me:

    The original motivation behind condensed infinity-groupoids is to create well-behaved categories of mathematical structures such as condensed HZ-module spectra in which one could do derived analytic geometry using category-theoretic methods without resorting to not-so-well behaved categories of topological spaces.

    In one reading this says that condensed ∞\infty-groupoids have be introduced to define condensed module spectra, which is a circular or empty statement. The next sentence gets closer to the point with the mentioning of analytic geometry, but it remains unclear what the poor topological spaces are being blamed for.

    The next sentence is alluding to what we once optimistically called condensed cohesion, but which seems to be at most condensed local contractibility. I have added a pointer to that entry now.

    diff, v7, current

  7. Added the hypercompletion condition, which is necessary, as explained in the pyknotic paper by Barwick and Haine. Also the simpler description using extremally disconnected sets.

    diff, v8, current

1 to 7 of 7

  1.