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A Reedy category $R$ is a special kind of category for which, given a model category $M$, we can define the Reedy model structure on the functor category $M^R$. I am trying to understand the nature of cofibrant objects under this model structure.

Given a Reedy diagram $X:R \rightarrow C$, where C is a model category, the nCatLab page on the Reedy model structure states that an object is cofibrant if and only if each map $L_rX \rightarrow X_r$ is a cofibration, where $L_rX$ is the latching object:

Given a diagram $X:R \rightarrow C$ and an object $r \in R$, it's latching object is $$ L_rX = \text{colim}_{s \rightarrow r} X_s,$$ where the colim is over the full subcategory of $R_+/r$ containing all objects except the identity $1_r$.

where $R_+/r$ denotes the slice category.

I wish to know where my following understanding is going wrong:

The diagram $*\rightrightarrows*$ is a Reedy diagram, where $*$ denotes the one point space.

The latching object for the first of the one point spaces is $\emptyset$, and for the second object it is $*$. Both the inclusion of $\emptyset$ into $*$ and of $*$ into $*$ are cofibrations, so that this diagram is cofibrant in the Reedy model structure.

Since in this case we have $R_+=R$, the projective model structure and Reedy model structure coincide. This would imply that the homotopy colimit and the colimit of this diagram coincide. However, they do not, since the homotopy colimit is $S^1$. Therefore something is wrong.

The revelance of this for me is that I am looking at Reedy diagrams of topological spaces where every map is a cofibrant inclusion, and I wish to understand which of these are cofibrant in the Reedy model structure in order to see in which cases the homotopy colimit and colimit coincide (as they do when the diagram is cofibrant).

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Denote the indexing category for your diagram by $f,g:a\rightrightarrows b$. You've incorrectly calculated $R_+/b$. Its objects are $f$ and $g$, and there are no non-identity morphisms, since neither of $f$ and $g$ factors through the other. Therefore the latching object for your diagram is the two-point space $*\sqcup *$. This explains why the homotopy colimit of your diagram is the colimit of $*\rightrightarrows I$, where the arrows are the inclusions of the two endpoints.

Other than the mistaken calculation, you seem to be assuming that the colimit of a Reedy cofibrant diagram is a homotopy colimit, but this is not true in general. Rather, the colimit of a projective cofibrant diagram is always a homotopy colimit. For direct categories like yours, the Reedy and projective model structures coincide, roughly since matching objects are trivial, but this isn't the case for a general Reedy category such as $\Delta^\mathrm{op}$.

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  • $\begingroup$ Thank you for another informative answer. I'll edit the question to clear up your second point; I am indeed aware (due to your answer to my previous question!) that the colimit of a projective cofibrant diagram is a homotopy colimit, and that in this case the two things coincide due to $R_+ = R$. $\endgroup$
    – Matt
    Jun 17, 2019 at 2:30

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