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nLab > Latest Changes: excisive functor

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2016

    have made explicit the proof that reduced excisive functors are equivalently spectra, here.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 8th 2016

    Any idea what the reduced nn-excisive functors will be? In general, I guess it’s messy. A while ago, I think I found:

    From this paper, it seems that J * 2(∞Grpd)J^2_{\ast}(\infty Grpd) has as objects pairs {A 1,A 2}\{A_1, A_2\}, where A 1A_1 is a spectrum and A 2A_2 is a spectrum with action of the symmetric group Σ 2\Sigma_2, and some map relating A 1A_1 and A 2A_2 via the Tate construction.

    But does anything pleasant happen as n→∞n \to \infty? I guess in the non-reduced case that’s just asking what H[X *]\mathbf{H}[X_\ast] looks like.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2016

    I have added statement and proof of excisive functors – Characterization via a generic stable object. Thanks to Eric Finster for hints.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 8th 2016
    • (edited Jan 8th 2016)

    As I said here, I don’t see why you’re calling

    H[X *]≃[(∞Grpd fin */) op,H] \mathbf{H}[X_\ast] \simeq [(\infty Grpd_{fin}^{\ast/})^\op, \mathbf{H}]

    the classifying (∞,1)-topos for pointed objects and not [(∞Grpd fin */) op,∞Grpd][(\infty Grpd_{fin}^{\ast/})^\op, \infty Grpd].

    Wild thought of the day: if H[X *]\mathbf{H}[X_\ast] is a kind of infinite jet topos, J ∞(H)J^{\infty}( \mathbf{H}), is there an (∞,2)(\infty, 2)-version of the jet comonad acting on (∞,1)Topos/H(\infty, 1)Topos/\mathbf{H}?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2016
    • (edited Jan 8th 2016)

    It’s the classifying H\mathbf{H}-topos, if you wish (at least once the typo with the “op” is fixed). And that’s what we are interesting in! Or maybe I am misunderstanding your question

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 8th 2016

    It’s the classifying H\mathbf{H}-topos,

    I haven’t seen that terminology before. At classifying topos we just hear of geometric morphisms from EE to a classifying topos as giving the relevant structures in EE.

    So here we’re to think of morphisms f:K→H[X *]f: \mathbf{K} \to \mathbf{H}[X_\ast] as giving some pointed objects in K\mathbf{K} (but something to do with H\mathbf{H})?

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 9th 2016
    • (edited Jan 9th 2016)

    So you must just mean taking place over H\mathbf{H}, where H[X *]→H\mathbf{H}[X_\ast] \to \mathbf{H} is given by the value at *\ast? It might be worth a word on classifying toposes over toposes at classifying topos, if anyone has the time.

    Is there an alternative familiar description of this topos in the case ∞Grpd[X *]\infty Grpd[X_\ast]?

    And which notation is to be used, H[X •]\mathbf{H}[X_\bullet] or H[X *]\mathbf{H}[X_\ast]?

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 9th 2016
    • (edited Jan 9th 2016)

    … if anyone has the time.

    Silly me. It’s already there:

    Classifying toposes can also be defined over any base topos SS instead of SetSet. In that case “Grothendieck topos” is replaced by “bounded SS-topos.”

    Hmm, do we have anything on the latter term? I see there’s unbounded topos, And ’bounded topos’ redirects to bounded geometric morphism, so I’ll put that link in.

    Everything just works the same in the (∞,1)(\infty, 1) case, presumably. And what is there to say about the (∞,1)(\infty, 1)-version of geometric theories, universal models, and anything else at classifying topos?

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJan 9th 2016

    Note that since the only (∞,1)(\infty,1)-toposes we have a definition for are the Grothendieck ones, all (∞,1)(\infty,1)-geometric morphisms are bounded. I don’t think that “(∞,1)(\infty,1)-geometric logic” has yet been written down explicitly.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 9th 2016

    Is Lurie’s ’bounded ∞\infty-topos’ of sec. 3.1.7 of Spectral Algebraic Geometry meant as an ∞\infty version of a bounded topos?

    You blink, it changes. It’s now A.1.7 in the latest version.

    Regarding “(∞,1)(\infty,1)-geometric logic”, I take it that’s what Urs is aiming at with geometric homotopy type theory, after the g+ discussion.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 9th 2016

    There must be 101 opportunities for a categorical logician wanting to go (∞,1)(\infty, 1), as is clear from what I quote of Lurie here.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJan 9th 2016

    The (∞,1)(\infty,1)-geometric logic here has to consist only of things preserved by inverse image functors, so in particular no universes and no function types. This is why I object to using geometric homotopy type theory for a type theory that does include those things.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJan 9th 2016

    And no, Lurie’s notion of “bounded” is, as far as I can tell, completely different. Arrgh.

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 9th 2016

    Then this might further upset you

    The terminology of Definition A.1.7.26 is slightly abusive. Any ∞- topos X can be regarded as an ∞-pretopos (Example A.1.6.5), but the condition that X is bounded as an ∞-topos (in the sense of Definition A.1.7.2) is not equivalent to the condition that X is bounded as an ∞-pretopos (in the sense of Definition A.1.7.26).

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 10th 2016
    • (edited Jan 10th 2016)

    Back to my ’wild thought’ from above

    if H[X *]\mathbf{H}[X_\ast] is a kind of infinite jet topos, J ∞(H)J^{\infty}( \mathbf{H}), is there an (∞,2)(\infty, 2)-version of the jet comonad acting on (∞,1)Topos/H(\infty, 1)Topos/\mathbf{H}?

    Would I be right in thinking that it’s possible to form a relative (∞,1)(\infty, 1)-tangent/jet of one (∞,1)(\infty, 1)-topos over another?

    Hmm, so according to the analogy with

    T infX× X(−)⊣Jet, T_{inf}X \times_X (-) \;\dashv\; Jet,

    from here, given two toposes over H\mathbf{H}, we might look for a left adjoint to such a relative Jet construction.

    Is H[X *]\mathbf{H}[X_\ast] then appearing in the case where the second of these toposes over H\mathbf{H} is H\mathbf{H} itself? So we’re seeing interest in [K,[∞Grpd fin */,H] H[\mathbf{K}, [\infty Grpd_{fin}^{\ast/}, \mathbf{H}]_{\mathbf{H}} as [K,JetH] H[\mathbf{K}, Jet \mathbf{H}]_{\mathbf{H}}, and this should be the same as [∞Grpd fin */×K,H] H[\infty Grpd_{fin}^{\ast/} \times \mathbf{K}, \mathbf{H}]_{\mathbf{H}}?

    But then why would ∞Grpd fin */×K\infty Grpd_{fin}^{\ast/} \times \mathbf{K} be the analogue of T infX× X(−)T_{inf}X \times_X (-)?

    Of course, what would help is an analogue of the de Rham space object ℑ(X)\Im(X) in some crazy differential cohesive (∞,2)(\infty, 2)-topos setting.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2016

    @David, sorry if that was unclear. It does say in the entry that H\mathbf{H} is used as the base topos, I thought that makes it clear.

    @Mike, I have no intention to fight for this terminology if you’d like to erase it. I just note that Steve Vickers in “Locales and toposes as spaces” (pdf) considers the same move, of boosting the terminology for all the flavors of logic (geometric, coherent, …) to the respective type theories. It seems the most obvious thing to do, and in any case some terminology or other seem to be needed here.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeFeb 10th 2016
    • (edited Feb 10th 2016)

    I have added a remark amplifying how the concept of excisive ∞\infty-functor may be regarded as fixing the failure of the concept of generalized homology functors to be strictly equivalent to spectra (instead of just receiving a map from them).

    The traditional definition of a generalized homology theory is as a functor on (finite) homotopy types with values in graded abelian groups. The Brown representability theorem says that these all arise from spectra EE via taking stable homotopy groups of smash products: X↦E •(X)≔π •(E∧X)X \mapsto E_\bullet(X) \coloneqq \pi_\bullet(E \wedge X). But due to the existence of phantom maps, this does not quite yield an equivalence between spectra and generalized homology theories.

    In view of this, the above proof may be thought of saying that this mismatch is fixed by refining homotopy groups by full homotopy types X↦Ω ∞(E∧X)X \mapsto \Omega^\infty(E\wedge X).

    I have added an analogous remark in the Idea section of phantom map.

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