When does the constant diagram functor preserve fibrant objects in the injective model structure on diagram categories?

When does the constant diagram functor preserve fibrant objects in the injective model structure on diagram categories?

For example, this is the case when the index category of the diagrams is a cofinite filtered poset (see Edwards and Hastings "Čech and Steenrod Homotopy Theories with Applications to Geometric Topology", LNM volume 542). But I would like some more general contexts.

• I only know this is the case in the context considered by Edwards and Hastings. – Eduardo J. Dubuc Sep 25 '17 at 20:36
• I added [ct.category-theory], to give this a top-level tag. – David Roberts Sep 25 '17 at 22:06
• If the indexing diagram is an "inverse category" i.e. a Reedy category where every morphism lowers degree, then this should hold. That's only mildly more general than what you've written though. – Dylan Wilson Sep 26 '17 at 1:57
• Over a Reedy category $R$, the injective and Reedy model structures often coincide -- e.g. if $R$ is elegant and the diagrams are in a topos, or if $R$ is skeletal (I'm not clear on the conditions on the codomain). The condition that the constant diagram functor over a Reedy category be Quillen actually appears as a hypothesis rather than a conclusion in several theorems here -- perhaps Riehl and Verity would know conditions guaranteeing this. – Tim Campion Sep 27 '17 at 23:11
• try to comment on this if you get a chance. – Mikhail Katz Oct 18 '17 at 9:45

1 Answer

This is a long comment, rather than an answer. The poset in Edward and Hastings context LNM 542 is elegant reedy, so the injective and the reedy model structures coincide, but independently of this, the constant diagram functor does not preserve fibrations, contrary to the statement in theorem 3.2.4. Consider the poset $A = \{\bullet \to \bullet \gets \bullet \}$, and a fibrant object $X \to 1$ in $\mathcal{C}$. It follows from 3.2.7 that if the constant diagram were a fibration in $\mathcal{C}^A$, then the diagonal $X \to X \times X$ must be a fibration in $\mathcal{C}$. However EH apply this to posets $P$ which are reindexing of filtering categories, so we add the following to include this situation. It is easy to see that this poset is the category $A = P_{< j}$ for the subdiagram $j = \{\bullet \to \bullet\}$ of two consecutive numbers in $\omega$, where $P$ is the result of the "Mardesic trick" (i.e. Deligne construction Prop. 8.1.6 SGA4) applied to $\omega$. Finally, it is easy to see directly that the constant diagram determined by a fibration $X \to Y$ is a fibration in $\mathcal{C}^P$ provided that for any $j \in P$, the finite poset $P_{< j}$ is either empty or connected, which clearly is not the case in the example above.