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 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 limit 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 simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject 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).
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 29th 2012
    • (edited Jun 29th 2012)

    People here may be familiar with my preprint the universal simplicial bundle is a simplicial group (aka ’W is a monad’). With a bit of tweaking I’ve found the following result (contractible means having the point as a deformation retract using simplicial homotopies):

    Let TT be a Lawvere theory containing that of monoids (better, consider a Lawvere theory equipped with an inclusion Th(Mon)TTh(Mon) \hookrightarrow T, giving a specified monoid operation mm) and CC a finite product category. Then there is a monad W TW_T on sTalg(C)s T alg(C) that takes a simplicial TT-algebra AA in CC and returns a contractible simplicial TT-algebra W TAW_T A with the unit a monomorphism (i.e. containing AA). If the specified mm is left cancellative (e.g. it underlies a group operation), then the canonical action of AA on W TAW_T A given by the inclusion map is free. (In any case, the action is as free as the action of AA on itself is.)

    If we pass to the underlying simplicial object u(W TA)u(W_T A) in CC then there is an AA-equivariant weak homotopy equivalence WAu(W TA)WA \to u(W_T A) where WAWA is the usual bar construction for mm (covered in Moore’s treatment of the functor WW, but not in May’s later book). If mm underlies a group operation then this weak homotopy equivalence is an isomorphism, and the quotient u(W TA)/Au(W_T A)/A exists in any finite-product category CC and is given by WA/A=W¯AW A/A = \overline{W}A. More generally, one could consider the simplicial action category, or perhaps pass to the category of simplicial presheaves on CC, if CC itself doesn’t have enough colimits. (Or even take the nerve of the action category to get a bisimplicial object in CC and take the diagonal.)

    Thus we have a simplicial TT-algebra structure on the total space of the ’universal AA-bundle’, for whenever we can find an interpretation of ’universal’ and ’bundle’ that makes sense. I wonder if this is good for anything. Given the recent result by Thomas, Urs and Danny about \infty-groups in \infty-toposes being presented by groups in simplicial sheaves, and \infty-bundles being represented by simplicial bundles, it seems natural to consider whether this result can be extended to \infty-algebras for other Lawvere theories. Then my result above is really a statement about \infty-algebras. What that conjectured result is good for I do not know either, but it has a sort of cohomological feel to it.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2012
    • (edited Jun 29th 2012)

    What can you say in general about (the existence of) the quotient map WAWA/AW A \to W A / A?

    For AA a simplicial group, the single fact that makes the whole theory tick is that this map is a Kan fibration resolution of the point inclusion *W¯G* \to \bar W G. It would be interesting for non-groupal AA if you get an inner Kan fibration.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 29th 2012
    • (edited Jun 29th 2012)

    Perhaps we can think about this as finding a fibrant replacement for the canonical point *BA\ast \to \mathbf{B}A, where we take BA\mathbf{B}A to be some internal connected (,1)(\infty,1)-category… Or is all this just hiding basic results behind fancy words?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2012

    Perhaps we can think about this as finding a fibrant replacement for the canonical point *→BA, where we take BA to be some internal connected (∞,1)-category…

    Yes, that’s what I mean.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 29th 2012

    Our comments crossed, Urs; I didn’t see what you had written when I posted my second one :-) I’ll reply in more detail later.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2012

    Oh, I see. :-)

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 29th 2012

    On second thoughts, taking the quotient by the monoid action is the wrong ging to do. I’m not sure what the right thing to do is, yet.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2012

    The lax quotient.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 30th 2012

    @Urs, Are you thinking of the diagonal of the nerve of the action simplicial category, or some other sort of quotient?

    I may have over-reacted about the monoid action. The whole point of W TW_T is to get something on which one can take an ordinary quotient, rather than some sort of homotopy quotient. Let me think some more before having a brain-vomit.