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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 18th 2018

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

• 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 $f$, let $f = gh$ and $h f^{-1} = g' h'$ be the unique factorizations. Then $id = ghf^{−1} = (gg')h'$, so $h' = id$ and $gg' = id$, whence $g = id$ and $g' = id$ since $g, g' \in R_+$. Thus $f = h \in R_−$. The same argument applied to $f^{−1}$ shows that $f$ preserves the degree, hence $f = id$.

• 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 $f$, let $f = gh$ and $h f^{-1} = g' h'$ be the unique factorizations. Then $id = gh f^{−1} = (gg')h'$, so $h' = id$ and $gg' = id$, whence $g = id$ and $g' = id$ since $g, g' \in R_+$. Thus $f = h \in R_−$. The same argument applied to $f^{−1}$ shows that $f$ preserves the degree, hence $f = id$.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 2nd 2020

For checking code for the $n$Lab best to use the Sandbox page.

The parser here on the $n$Forum does not behave identically to that for $n$Lab pages.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeJan 31st 2021

Redirect: latching map, matching map.

• CommentRowNumber6.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 8th 2021

Monoidal Reedy model structures are discussed in

• CommentRowNumber7.
• CommentAuthormbid
• CommentTimeOct 21st 2021

Add characterization of when diagonal functors $C -> C^R$ are left/right Quillen.

• CommentRowNumber8.
• CommentAuthorThomas Holder
• CommentTimeMar 24th 2023

• 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. )

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 19th 2023

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

In trying to see this in detail, I am looking — given a morphism $f_\bullet \colon X_\bullet \to Y_\bullet$ in $s\mathcal{A}$ — at the diagrams

$\array{ L_r X &\longrightarrow& X_r \\ \big\downarrow && \big\downarrow \\ L_r Y & \longrightarrow & L_r Y \overset{L_r X}{\sqcup} X_r & \longrightarrow & Y_r \\ \big\downarrow && \big\downarrow && \big\downarrow \\ 0 &\longrightarrow& X_r / L_r X &\longrightarrow& Y_r/L_r Y }$

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 $f_\bullet$ is Reedy cofibrant in that $L_r Y \overset{L_r X}{\sqcup} X_r \longrightarrow Y_r$ is a cofibration in $\mathcal{A}$ for each $r$

$\Rightarrow$ the induced morphism $X_r / L_r X \longrightarrow Y_r/L_r Y$ is a cofibration in $\mathcal{A}$, which is the $r$th 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 $L_r X \to X$ 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.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeApr 20th 2023

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

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeApr 20th 2023

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

• 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.

• 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.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeMay 9th 2023

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