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I am new to MO and I hope that this question is suitable for it. Following Hovey's "Model Categories" ( http://ericmalm.net/ac/projects/symmetric-spectra/hovey--model-cats.pdf ) to study model structures on functor categories, it seems that something is missing, or better I proved something to proceed further, but I am not really sure about this result.

It seems to me that, considering a model category $\mathscr{C}$ and the simplicial category $\Delta$ and endowing $\mathscr{C}^{\Delta}$ with the Reedy model structure, there is a natural isomorphism $L_n \ l^{\bullet}A \simeq l^{\bullet}A [n] \ \ \forall A \in \mathscr{C}$.

Let me explain a bit the notation (but everything can be found on pages 124 to 127 in the previous link): by $L_n$ is meant the $n$-th latching space functor ( http://ncatlab.org/nlab/show/Reedy+model+structure) and $l^{\bullet}A$ is the cosimplicial object in $\mathscr{C}$ whose $n$-th "space" is the $n+1$-fold coproduct of $A$, with the obvious cosimplicial structure. Furthermore, it is left adjoint to the evaluation functor $Ev_0 : \mathscr{C}^{\Delta} \to \mathscr{C}$

The abovementioned isomorphism comes out by unraveling definitions and using universal properties of colimits and adjointness, at least for what I have tried.

Could someone confirm this fact, or disprove it (so that I have to find another way to prove the claim I am interested into)?

Thanks in advance, any help will be highly appreciated.

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  • $\begingroup$ The latching object at $1$ is the initial object, not $A$, it seems to me. $\endgroup$ Sep 5, 2014 at 12:02
  • $\begingroup$ Do you mean $L_n \ l^{\bullet} 1 \simeq 0 \ \forall n \in \Delta$? $\endgroup$ Sep 5, 2014 at 12:15
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    $\begingroup$ Indeed, the latching object at the zero-simplex is the initial object, not $A$. On the other hand, at all other simplices, it suffices to see that your cosimplicial object takes the $n$-simplex $[n]$ to $S([n]) \otimes A$. Here, $S = \Delta([0],-) :\Delta \to \mathrm{Set}$ is a functor which takes $[n]$ to the set $\{0,\ldots,n\}$, and $X \otimes A$ is the $X$-fold coproduct of copies of $A$ (i.e. the cotensor with the set $X$). If $C$ has coproducts, the functor $-\otimes A: \mathrm{Set} \to C$ is left adjoint to $C(A,−)$, so it preserves all colimits. (to be continued...) $\endgroup$ Sep 5, 2014 at 16:07
  • $\begingroup$ (continuation) Consequently, it suffices to calculate the latching object of the functor $S : \Delta \to \mathrm{Set}$ itself. It is not difficult to check that for all $n>0$, the latching object of $S$ at $[n]$ is $S([n])$ itself. On the other hand, the latching object of $S$ at $[0]$ is empty. $\endgroup$ Sep 5, 2014 at 16:08
  • $\begingroup$ Thank you! This is very helpful because it seems to solve two problems at once.. Indeed you have proven that $L_0$ of something is always $0$ (I knew this, but I didn't remember this fact while I was writing my question), but that my iso holds in degree $n>0$, which is what I needed. $\endgroup$ Sep 5, 2014 at 17:11

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