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nLab > Latest Changes: model structure on functors

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2010

    expanded model structure on functors by adding a long list of properties

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2012
    • (edited Mar 17th 2012)

    I have edited the Properties-section at model structure on functors to make more explicit that all of the statements about functorial dependen on domain and codomain hold also for the case of S\mathbf{S}-enriched functors on S\mathbf{S}-enriched categories.

    (This was previously mentioned in between the lines, but not made sufficiently explicit.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2016
    • (edited Apr 13th 2016)

    I have added pointer to

    • Robert Piacenza, Homotopy theory of diagrams and CW-complexes over a category, Can. J. Math. Vol 43 (4), 1991 (pdf)

      also chapter VI of Peter May et al., Equivariant homotopy and cohomology theory, 1996 (pdf)

    to the entries model structure on functors, Elmendorf’s theorem and Model categories of diagram spectra.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeDec 15th 2018

    Why does HTT Proposition A.3.3.2 (existence of injective and projective model structures on enriched diagram categories) require S\mathbf{S} to be excellent? In particular, why does it require the “invertibility hypothesis”? The proof isn’t written out but said to be “identical to that of Proposition A.2.8.2”, the existence of injective and projective model structures on unenriched diagram categories; I glanced at the latter proof but wasn’t able to see where the invertibility hypothesis might be used.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJan 1st 2019

    Uniformized the notation (D = small category, C = model category), separated the definition of the potential model structures from the theorems about their existence, and included some alternative existence theorems that don’t require cofibrant generation.

    diff, v40, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJan 17th 2019

    Added Moser’s theorem about existence of both projective and injective model structures for all accessible model categories, which is much more general than Lurie’s (although it doesn’t imply that the injective model structure is cofibrantly generated, only that it is accessible).

    diff, v42, current

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 13th 2019

    Replaced Lurie’s theorem on the existence of injective model structures with one derivable from Makkai-Rosicky that doesn’t need the enriching category to be “excellent”.

    diff, v43, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    moved this proposition (on relation to functor ∞\infty-categories) out of the subsection “Properties – Functoriality in domain and codomain” (where it was misplaced, into “Properties – General”).

    Also adjusted typesetting of arrows in the entry (replacing \leftarrow by \longleftarrow and \to by \longrightarrow in order for the adjunctions to display properly)

    diff, v50, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2023

    Where it said “cofibrantly generated and hence combinatorial” (here) I have added after “hence” a pointer to the relevant Proposition.

    diff, v52, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2023
    • (edited May 14th 2023)

    added (here) a brief paragraph stating the relation to ∞\infty-functors (Prop. 4.2.4.4 in HTT).

    N(sFunc(D,C) ∘)≃qCatFunc ∞(N(D),N(C ∘)). N \big( \mathbf{sFunc}(\mathbf{D}, \mathbf{C})^\circ \big) \;\; \underset{qCat}{\simeq} \;\; Func_\infty \Big( N(\mathbf{D}) ,\, N\big( \mathbf{C}^\circ \big) \Big) \,.

    (We have corresponding and more discussion at (infinity,1)-category of (infinity,1)-functors, but it’s good to have a pointer here, too).

    By the way: Is there a general naturality statement saying that this equivalence is suitably natural, e.g. compatible with precomposition operations f *f^\ast?

    [edit: Have posted what looks like an answer to this question in another thread: here]

    diff, v59, current

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 14th 2023

    Re #10: There is a question about this on MathOverflow: https://mathoverflow.net/questions/440965/htt-remark-4-2-4-5

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2023

    Thanks, I have added a footnote: there.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2023

    Added more explicit pointer to Hirschhorn where the existence of the projective model structure is stated (here).

    diff, v60, current

1 to 13 of 13

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