Not signed in (Sign In)

Start a new discussion

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 beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory 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 galois-theory 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 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 string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2021

    starting something, meaning to record a transparent proof of an adjoint equivalence that becomes a Quillen equivalence for simplicial presheaves (haven’t seen this discussed anywhere).

    But not done yet, just need to save…

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2021

    I have now typed out full proof of the adjoint equivalence.

    It follows (not in the entry yet) with K. Brown’s lemma that for simplicial presheaves this adjoint equivalence is a Quillen adjunction for the projective model structure and the slice of that.

    Still need to think about full proof that this is a Quillen equivalence. But need to interrupt now.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2021

    Have typed out the argument (here) that the simplicial version of the adjoint equivalence is a Quillen adjunction.

    Still need to show that it’s a Quillen equivalence.

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2021

    Finally I have typed out also the proof of the Quillen equivalence property (here)

    diff, v9, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2021

    Have now added a concluding section (here) with the statement seen in \infty-category theory.

    Also adjusted the Idea-section, to reflect this completion of the proofs.

    diff, v10, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2021

    Added mentioning (here) of the application to the proof of cohesion of global- over G-equivariant homotopy theory.

    diff, v10, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    Following discussion in another thread (here)…

    …I have added (here) a section which proves the equivalence of categories in the generality where the base object need not be representable.

    Strictly speaking, the statement proven in this case is (currently) a little weaker than that given over a representable, (which proves an adjoint equivalence with identification of the right adjoint as the functor of sections). But the proof is general abstract and applies verbatim also in \infty-category theory (where all the required properties, such as the interplay of hom-functors with colimits, still hold as natural equivalences of \infty-groupoids).

    I have added a brief remark on this here.

    diff, v11, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2022

    I have slightly adjusted and re-arranged throughout the entry, for better readability (hopefully).

    diff, v13, current

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 10th 2022


    In simplicial presheaf theory


    We now promote the Quillen equivalence in the previous section to the case of Čech-[[local model structures on simplicial presheaves].

    Recall that these are obtained as a left Bousfield localization of the (say) projective model structure on simplicial presheaves with respect to Čech nerves of covering families.

    We reuse the notation of the previous section.

    \begin{proposition} The Quillen equivalence of \ref{SimplicialLocalSectionsIsRightQuillen} descends to a Quillen equivalence of the corresponding Čech-local projective model structures. \begin{tikzcd} \big( \mathrm{sPSh}(\mathcal{C}){/y{\mathcal{C}}(X)} \big){\mathrm{\v Cech,proj}} \ar[ rrr, shift right=8pt, “{ \mathrm{PSh}(\mathcal{C}){/y(X)} \big( (y_{\mathcal{C}}){/X}(-) ,, -
    \big) }”{below} ] &&& \mathrm{sPSh} \big( \mathcal{C}
    {/X} \big)_{\mathrm{\v Cech,proj}} \ar[ lll, shift right=8pt ] \ar[ lll, phantom, “{ \scalebox{.6}{ Qu\simeq_{\mathrlap{\mathrm{Qu}}}} }” ] \end{tikzcd} \end{proposition}

    \begin{proof} Both model categories are left proper and combinatorial. Therefore we can take left Bousfield localizations with respect to arbitrary sets of morphisms.

    We localize both sides with respect to Čech nerves of respective covering families. Observe that Čech nerves of covering families in 𝒞 /X\mathcal{C}_{/X} are mapped to Čech nerves of covering families in sPSh(𝒞)sPSh(\mathcal{C}) and therefore also in the slice category sPSh(𝒞) y 𝒞(X)sPSh(\mathcal{C})_{y_{\mathcal{C}}(X)}. Thus, we have an induced Quillen adjunction between localized model categories.

    It remains to show that this Quillen adjunction is a Quillen equivalence.

    It suffices to show that the right adjoint reflects weak equivalences between fibrant objects. Here fibrant objects are objectwise Kan complexes that satisfy the appropriate variant of the homotopy descent property. Local weak equivalences between locally fibrant objects coincide with objectwise weak equivalences. As established in the previous section, the right adjoint functor reflects objectwise weak equivalences between objectwise fibrant presheaves, which completes the proof. \end{proof}

    diff, v14, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)