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nLab > Latest Changes: simplicial object

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2010

    added to simplicial object a section on the canonical simplicial enrichment and tensoring of D Δ opD^{\Delta^{op}} for DD having colimits and limits.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeMar 22nd 2012

    I just noticed this in simplicial object as the Idea:

    A simplicial object XX in a category CC is a collection {X n} n∈ℕ\{X_n\}_{n \in \mathbb{N}} of objects in CC that behave as if X nX_n were an nn-dimensional simplex internal to CC.

    As someone who works a lot with simplicial objects this does not correspond to my idea of a simplicial object and quite frankly I do not understand the meaning of ‘simplex internal to CC’. Does anyone else feel like I do? I think that a couple of instances of simplcial objects would give the idea better.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2012

    I have modified the sentence. But if anyone has more leisure for it than I currently do, feel free to have a go at it.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeMar 23rd 2012

    Thanks that is already much better. I have added in a bit more. I am wondering if there is s specific example that is general enough to be ’of use’ for a large number of people (other than simplicial sets). I will add one slightly later if I can.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2012

    The entry lists a bunch of Examples. Are you looking for something else?

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeMar 23rd 2012

    I am wondering if there should be a specific example (e.g. the smooth singular complex of a Lie group can be considered in several settings as giving a simplicial group, simplicial topological group, etc. but that example fits better elsewhere). It may not be needed as there are other examples in other entries. Perhaps ‘If you want a specific example of this type of construction look at …’ might be worth putting somewhere. I am in two minds about it.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJan 23rd 2019

    Add a remark about geometric realization and the fact that it is a simplicial functor.

    diff, v30, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2020

    added ISBN to

    diff, v31, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2021

    added pointer to:

    • Alexander Grothendieck, p. 108 (11 of 21) in: Techniques de construction et théorèmes d’existence en géométrie algébrique III : préschémas quotients, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf)

    diff, v34, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 23rd 2022

    Added:

    The original definition of simplicial objects, maps between them, and homotopies of such maps is due to Daniel M. Kan:

    • Daniel M. Kan, On the homotopy relation for c.s.s. maps, Boletín de la Sociedad Matemática Mexicana 2 (1957), 75–81. PDF.

    diff, v38, current

1 to 10 of 10

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