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nLab > Latest Changes: Gamma-space

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  1.  
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 23rd 2012

    Created Gamma-space.

    • CommentRowNumber2.
    • CommentAuthorMatanP
    • CommentTimeFeb 25th 2012
    I don't feel comfortable enough to edit, but there are two references that fit here:

    "Categories and cohomology theories", Topology 13 - Segal - 1974

    "Homotopy theory of Γ-spaces, spectra, and bisimplicial sets”, Geometric applications of homotopy theory - Bousfield, Friedlander - 1977

    in the first, there are also "special delta spaces" that correspond to 1-fold loop spaces and in

    C. Balteanu, Z. Fiedorowicz, R. Schwanzl and R. Vogt, "Iterated
    monoidal categories", Advances in Mathematics 176 pp. 277–349
    (2003).

    there is the n-fold generalization of "special Delta spaces" which are n-simplicial spaces (i.e. functors X:(Delta^op)^n--->spaces) that satisfy the analogous Segal condition. These correspond to n-fold loop spaces if \pi_0 X(1,...,1) is a group.
    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 25th 2012
    • (edited Feb 25th 2012)

    Hi there Matan. Editing is easy! Open the page, go to Edit. (and if you want to add a reference see if that reference is somewhere else in the lab! That makes life easy if it .. copy from the source of that page into where you want the reference to go.)

    have a look at the source and … do the obvious things. If it goes wrong go back to the previous version and call for help!!!!!

    (Some seconds later: I was going to give you a hand but note that you are actually editing that page anyway….. )

    One point to add would be to make the names point to the relevant page entries, also any pages you know exist can be linked so G. Segal should work but did not (but I did a redirect).

    • CommentRowNumber4.
    • CommentAuthorjim_stasheff
    • CommentTimeFeb 25th 2012
    How about adding or pointing to Theta_n spaces/objects?
    • CommentRowNumber5.
    • CommentAuthorMatanP
    • CommentTimeFeb 25th 2012
    Hi Tim.
    Thanks for the welcoming. I'll try editing some more. I want to add a new entry on special delta spaces; how do I do it?
    BTW, do you know of a better name for special delta spaces?

    Jim: do you mean algebraic theories (e.g. Badzioch http://www.jstor.org/pss/3062135) or something else I'm not aware of?
    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 25th 2012
    • (edited Feb 25th 2012)

    I see you solved the mystery!!! My methods are either to put in the link in an existing (relevant) entry, or to search on the term and if it does not exist and is useful, the search page suggests that you create it.

    It is always better to search first (and with possible alternative titles) as that we we don’t get too many needs to merge entries. Clearly someone else may have used a slightly different term for something or it may be hidden in another entry. If a mention is hidden in another entry it is a good thing to go in to that and convert the term to a link.

    I wonder if jim is meaning Theta-space? They certainly look to be related but am not up on those as much as I should be.

    It would be good if you added a bit more about your research interests on your own page.

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeDec 11th 2014

    The introduction seems to be misleading: a grouplike Γ\Gamma-space is an infinite loop space, but are Γ\Gamma-spaces automatically grouplike? It appears to me, for instance, that any commutative monoid gives rise to a Γ\Gamma-space.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2014

    Thanks for the alert, I have fixed the wording.

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 1st 2020

    Redirects.

    diff, v18, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 18th 2021

    Added:

    Delooping

    One of the main advantages of Γ\Gamma-spaces (and, more generally, Γ\Gamma-objects) is that the delooping construction is very easy to express in this language.

    The delooping construction is a functor

    B:Fun(Γ op,M)→Fun(Γ op,M),B\colon Fun(\Gamma^{op},M) \to Fun(\Gamma^{op},M),

    where MM is the relative category for which we are considering Γ\Gamma-objects. The most common choices are M=sSetM=sSet, the model category of simplicial sets, and M=TopM=Top, the model category of topological spaces.

    We define

    (BA)(S):=hocolim(T↦A(S×T)),(B A)(S) := hocolim (T\mapsto A(S\times T)),

    where T∈Δ opT\in\Delta^\op and the argument of the homotopy colimit functor is a simplicial object in MM. Here T∈ΔT\in\Delta is converted first to an object of Γ\Gamma via the functor Δ→Γ\Delta\to\Gamma described below.

    diff, v19, current

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 18th 2021

    Added:

    Another early reference considers Γ\Gamma-objects in simplicial groups. It is also the first reference that uses the terms “special Γ-spaces” and “very special Γ-spaces”, which it attributes to Segal.

    • Donald W. Anderson, Chain functors and homology theories, Symposium on Algebraic Topology, Lecture Notes in Mathematics (1971), 1–12. doi.

    diff, v20, current

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeApr 16th 2024

    Link for pdf at

    • {#Lydakis98} Lydakis, Smash products and Γ\Gamma-spaces, Math. Proc. Cam. Phil. Soc. 126 (1999), 311-328 (pdf)

    does not seem to work.

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 16th 2024

    Re #12: Broken links are easy to repair using Web Archive.

    For example, the link in #12 is now https://web.archive.org/web/20210117115922/http://hopf.math.purdue.edu/Lydakis/smash_gamma.pdf

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2024

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