On the nlab


it is claimed that a simplicial object in a model category, in which all monomorphisms are cofibrations, is always Reedy-cofibrant.

Does anyone know a reference for this?

In the case of simplicial sets the statement is Theorem 15.8.7 in Hirschhorn's book.

Follow up question: Is it dually true that any cosimplicial object in a model category, in which all epimorphisms are fibrations, is Reedy-fibrant?

Again, is there a reference?

Thanks a lot.

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    $\begingroup$ There surely must be some condition on the model category to make this true. Its truth for sets implies its truth in any presheaf category, and hence in any Grothendieck topos, hence in any Cisinski model category. But I'd be surprised if it were true in any category at all. $\endgroup$ – Mike Shulman May 18 '18 at 17:41
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    $\begingroup$ Apparently that's my bad. I forget what I thought when I wrote that statement. Fixed it here: ncatlab.org/nlab/show/… $\endgroup$ – Urs Schreiber May 18 '18 at 19:14
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    $\begingroup$ Thank you for your comments. How about the (opposite) category of chain complexes? In the end, I want to conclude that every cosimplicial (unbounded) chain complex is Reedy fibrant. Here I equip chain complexes with the projective model structure. $\endgroup$ – Lukas Woike May 18 '18 at 19:49

The claim would be true if it were the case that for any simplicial object $X_\bullet$ in some (cocomplete) category $C$, the maps $L_nX\to X_n$ from the latching objects are always monomorphisms. This is the case when $C$ is any topos for instance. It is also true when $C$ is any additive category (because of the Dold-Kan theorem on (co)simplicial objects in an additive category with retracts), which would seem to include Faelivrin's example.

But it is not true generally. For instance, if $C$ is the category of augmented commutative $k$-algebras. Pick an augmented commutative $k$-algebra $A$, and form the simplicial object with $X_n= A\times_k\cdots \times_k A$ ($n$ copies of $A$); this is an example of a 1-coskeleton. The map $L_2X\to X_2$ has the form $$ A\otimes_k A \to A\times_k A, $$ and this can't generally be a monomorphism, e.g., when $A=k[x]$.

Probably an actually counterexample to the claim can be constructed from an idea like this.

  • $\begingroup$ Thanks a lot for this answer! Can you maybe explain further why the Dold-Kan theorem (you mean the Dold-Kan correspondence?) will prove the claim for an additive category? $\endgroup$ – Lukas Woike May 20 '18 at 17:12
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    $\begingroup$ @Faelvirin I guess I really mean the proof of the Dold-Kan correspondence, whch will explicitly provide a retraction to the map $L_nX\to X_n$. $\endgroup$ – Charles Rezk May 20 '18 at 19:45
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    $\begingroup$ Or more directly: if $A$ is your additive model category, and $C$ the class of cofibrations in it, let $sA$ be simplicial objects, and $C_R$ the reedy cofibrations. Then under $sA\approx Ch_{\geq0}(A)$, the class $C_R$ is taken exactly to the class of chain maps which are in $C$ in each degree. $\endgroup$ – Charles Rezk May 20 '18 at 19:59

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