On the nlab


<http://ncatlab.org/nlab/show/Reedy+model+structure#fibrant_and_cofibrant_objects>


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.



Reference request for statement on nlab: Reedy (co)fibrancy of (co)simplicial objects
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.


