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nLab > Latest Changes: Reedy model structure

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2018

    fixed the statement of Example 5.2 (this example) by restricting it to 𝒞=sSet\mathcal{C} = sSet

    diff, v36, current

    • CommentRowNumber2.
    • CommentAuthorYuxi Liu
    • CommentTimeJul 2nd 2020

    I’m just here to check the code since there’s no way to check in the editor.

    Proof: Take any isomorphism ff, let f=ghf = gh and hf −1=g′h′h f^{-1} = g' h' be the unique factorizations. Then id=ghf Unknown characterUnknown characterUnknown character1=(gg′)h′id = ghf^{−1} = (gg')h', so h′=idh' = id and gg′=idgg' = id, whence g=idg = id and g′=idg' = id since g,g′∈R +g, g' \in R_+. Thus f=h∈R Unknown characterUnknown characterUnknown characterf = h \in R_−. The same argument applied to f Unknown characterUnknown characterUnknown character1f^{−1} shows that ff preserves the degree, hence f=idf = id.

    • CommentRowNumber3.
    • CommentAuthorYuxi Liu
    • CommentTimeJul 2nd 2020

    I’m just here to check the code since there’s no way to check in the editor.

    Proof: Take any isomorphism ff, let f=ghf = gh and hf −1=g′h′h f^{-1} = g' h' be the unique factorizations. Then id=ghf Unknown characterUnknown characterUnknown character1=(gg′)h′id = gh f^{−1} = (gg')h', so h′=idh' = id and gg′=idgg' = id, whence g=idg = id and g′=idg' = id since g,g′∈R +g, g' \in R_+. Thus f=h∈R Unknown characterUnknown characterUnknown characterf = h \in R_−. The same argument applied to f Unknown characterUnknown characterUnknown character1f^{−1} shows that ff preserves the degree, hence f=idf = id.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2020

    For checking code for the nnLab best to use the Sandbox page.

    The parser here on the nnForum does not behave identically to that for nnLab pages.

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 31st 2021

    Redirect: latching map, matching map.

    diff, v38, current

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 8th 2021

    Added:

    Monoidal Reedy model structures are discussed in

    diff, v40, current

    • CommentRowNumber7.
    • CommentAuthormbid
    • CommentTimeOct 21st 2021

    Add characterization of when diagonal functors C−Unknown characterC RC -> C^R are left/right Quillen.

    diff, v41, current

    • CommentRowNumber8.
    • CommentAuthorThomas Holder
    • CommentTimeMar 24th 2023

    Added a reference to

    diff, v44, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2023

    I have re-organized and adjusted the section Definition – Plain version (here) to be more systematic.

    (It used to start out with a rather pointless “Theorem” stating just that “there exists a model structure with objectwise weak equivalences”, then claimed to explain the “basic idea” but instead incrementally provided the actual definitions. I have turned all that around, added Definition/Proposition-environments etc. and pointers to the literature. )

    diff, v46, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2023

    In this comment, Charles Rezk once said that for 𝒜\mathcal{A} an additive model category, the Reedy cofibrations in s𝒜s \mathcal{A} correspond, under the Dold-Kan correspondence, exactly to the degreewise cofibrations in Ch ≥0(𝒜)Ch_{\geq 0}(\mathcal{A}).

    In trying to see this in detail, I am looking — given a morphism f •:X •→Y •f_\bullet \colon X_\bullet \to Y_\bullet in s𝒜s\mathcal{A} — at the diagrams

    L rX ⟶ X r ↓ ↓ L rY ⟶ L rY⊔L rXX r ⟶ Y r ↓ ↓ ↓ 0 ⟶ X r/L rX ⟶ Y r/L rY \array{ L_r X &\longrightarrow& X_r \\ \big\downarrow && \big\downarrow \\ L_r Y & \longrightarrow & L_r Y \overset{L_r X}{\sqcup} X_r & \longrightarrow & Y_r \\ \big\downarrow && \big\downarrow && \big\downarrow \\ 0 &\longrightarrow& X_r / L_r X &\longrightarrow& Y_r/L_r Y }

    Here the top square, the left rectangle and the bottom rectagle are pushouts by definition, whence the pasting law implies first that the left bottom square and then that the right bottom square is a pushout.

    By pushout-stability of cofibrations, this yields that:

    If f •f_\bullet is Reedy cofibrant in that L rY⊔L rXX r⟶Y rL_r Y \overset{L_r X}{\sqcup} X_r \longrightarrow Y_r is a cofibration in 𝒜\mathcal{A} for each rr

    ⇒\Rightarrow the induced morphism X r/L rX⟶Y r/L rYX_r / L_r X \longrightarrow Y_r/L_r Y is a cofibration in 𝒜\mathcal{A}, which is the rrth component of the corresponding map of normalized chain complexes.

    This gives one direction of the statement. But how about the backwards implication “⇐\Leftarrow”?

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 19th 2023

    Re #10: The other implication makes use of the concrete structure of the Dold–Kan functor Γ.

    In detail, Γ will free degenerate elements in degree n along all possible degenerate simplicial maps.

    This means that the latching object of ΓC in degree n can be computed simply as the inclusion of (ΓC)_n without C_n into (ΓC)_n.

    This is an inclusion of a direct summand, therefore the latching map is a cofibration.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023

    This is the argument that L rX→XL_r X \to X is a monomorphism in abelian categories. But the claim referred to in #10 is stronger, or even different, isn’t it?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023

    For the record, I have made that simpler statement fully explict, here.

    diff, v48, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023

    added (here) that Reedy model structures induced from cofibrantly generated model categories are themselves cofibrantly generated

    diff, v51, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023

    added (here) the corollary that a Reedy model category with values in a combinatorial model category is itself combinatorial

    diff, v52, current

    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 20th 2023
    • (edited Apr 20th 2023)

    Re #12: It’s not just monomorphism, it’s a split monomorphism, as shown in the construction of the Dold–Kan functor.

    More precisely, the latching map for ΓC→ΓD will have the form L_n Y ⊔ C_n → L_n Y ⊔ D_n, so it is a cofibration as long as C_n → D_n is one, which it is by assumption.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2023

    Oh, right, sorry, I was being stupid. Have adjusted here.

    diff, v53, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2023

    added (here) the statement that the Reedy structure into a left proper combinatorial model category is itself (combinatorial and) left proper.

    diff, v54, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2023

    added the statement (here) that Reedy model structures inherit properness

    diff, v60, current

    • CommentRowNumber20.
    • CommentAuthoranuyts
    • CommentTimeFeb 9th 2024

    Specify that latching & matching objects live in C.

    diff, v63, current

    • CommentRowNumber21.
    • CommentAuthoranuyts
    • CommentTimeSep 9th 2024

    More intuition for latching & matching.

    diff, v65, current

1 to 21 of 21

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