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Consider the discrete combinatorial model structure on the category of $\Delta$-generated spaces: all maps are cofibrations and fibrations and the weak equivalences are the homeomorphisms. Applying Proposition A.3.2.4 of (Higher Topos Theory, Lurie) By mimicking the construction of the canonical model structure on the category of small categories, one obtains a model structure on the category of topologically enriched small categories and enriched functors such that the weak equivalences $F:C\to D$ are the local homeomorphisms such that the underlying functor $F_0:C_0\to D_0$ is essentially surjective.

Like in the non-enriched case, is this model structure unique (I mean for this class of weak equivalences) ? Is it "canonical" in this sense ?

EDIT: The discrete model structure on the category of $\Delta$-generated spaces is only accessible, not combinatorial because the underlying category is not locally finitely presentable.

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  • $\begingroup$ Just to be clear, by "local homeomorphism in $Cat_{Top_\Delta}$", you mean a functor $F$ which is fully faithful in the $Top_\Delta$-enriched sense that $Hom(a,b) \to Hom(Fa,Fb)$ is an isomorphism in $Top_\Delta$ (i.e. a homeomorphism) for all $a,b$. So this notion is unrelated to the notion of a local homeomorphism in $Top_\Delta$. Since this is potentially confusing, I think I'd prefer the term "$Top_\Delta$-fully-faithful" or something like that. So a weak equvialence in this "canonical" model structure is precisely a $Top_\Delta$-enriched equivalence of categories. $\endgroup$ – Tim Campion Sep 21 '18 at 19:37
  • $\begingroup$ @TimCampion I believed that that was the standard terminology; yes I meant that; And you're right, 'local' has not exactly the same meaning in Lurie's book. I'll be careful. $\endgroup$ – Philippe Gaucher Sep 22 '18 at 6:45
  • $\begingroup$ Oh, I agree it's the standard terminology, it's just that the standard terminology clashes $\endgroup$ – Tim Campion Sep 23 '18 at 1:01
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Yes, the canonical model structure is unique. The uniqueness of the canonical model structure on $Cat$ was nicely exposited by Chris Schommer-Pries on the Secret Blogging Seminar back in the day. Let's go through and mimic the proof there.

Explicit description of the canonical model structure:

Let $(C,W,F)$ denote the cofibrations, weak equivalences, and fibrations of the canonical model structure. The cofibrations are generated by $\emptyset \to pt$, along with $\Sigma X \to \Sigma Y$ for every map $X \to Y$ in $Top_\Delta$, where $\Sigma X$ is the $Top_\Delta$-enriched category with two objects $0,1$ and $Hom(0,0) = Hom(1,1) = pt$, $Hom(0,1) = X$, and $Hom(1,0) = \emptyset$. Therefore

  • $C$ consists of the injective-on-objects enriched functors.

  • $C \cap W$ consists of the injective-on-objects enriched equivalences. So it is generated by an inclusion $pt \to E$ where $E$ is the walking isomorphism.

  • $W$ consists of the enriched equivalences.

  • $W \cap F$ consists of the surjective-on-objects enriched equivalences.

  • $F$ consists of the isofibrations.

Uniqueness of this model structure:

Let $(C',W,F')$ be the cofibrations, weak equivalences, and fibrations of a model structure on $Cat_{Top_\Delta}$ with the same weak equivalences as the canonical model structure. We'll use "cofibration", "$\hookrightarrow$", "fibration", and $\twoheadrightarrow$ in the sense of this model structure, and "canonical cofibration" and "canonical fibration" to refer to the notions from the canonical model structure.

  1. $\emptyset \hookrightarrow pt \in C'$.

    For there must be some nonempty cofibrant object $X$, and then $0 \to pt$ is a retract of $\emptyset \hookrightarrow X$.

  2. $C' \supseteq C$, $C' \cap W \supseteq C \cap W$, $F' \cap W \subseteq F \cap W$, $F' \subseteq F$.

    For since $\emptyset \to pt \in C'$, we have that every $ f \in F' \cap W$ is surjective on objects. Since $f$ is also an enriched equivalence. we have $f \in F \cap W$, i.e. $F' \cap W \subseteq F \cap W$. The other inclusions follow formally.

  3. If $C' \neq C$, then $E \to pt \in C' \cap W$ (where again $E = (0 \cong 1)$ is the walking isomorphism).

    If $C' \neq C$, then by (2), $C' \not \subseteq C$, so there is a cofibration $A \hookrightarrow B \in C' \setminus C$ which is not injective on objects. Pick $x,y \in A$ which map to the same object in $B$. There is a unique enriched functor $A \to E$ sending $x$ to 0 and $y$ to 1. Then by pushout we obtain a cofibration $E \hookrightarrow E \cup_A B$ sending 0 and 1 to the same point. The codiscretification map $E \cup_A B \hookrightarrow (E\cup_A B)^\flat$ is injective on objects, i.e. a canonical cofibration, and hence a cofibration. Composing, we obtain a cofibration $E \hookrightarrow E \cup_A B \hookrightarrow (E\cup_A B)^\flat$ sending 0 and 1 to the same point, and sending the isomorphism between them to the identity. So $E \hookrightarrow pt$ is a retract of this map and hence also a cofibration. Since $E \to pt \in W$ is an enriched equivalence of categories, the claim follows.

  4. If $C \neq C'$, then every fibrant object $X$ has no non-identity isomorphisms.

    $X \to pt$ must lift against $E \to pt$ by (3).

  5. $C = C'$, i.e. the model structures agree.

    Every object must be equivalent to a fibrant object. But enriched equivalences preserve the property of having objects with non-identity automorphisms. So because there exist $Top_\Delta$-enriched categories with objects with non-identity automorphisms, this contradicts (4) if $C \neq C'$.


This argument generalizes to any cartesian enriching category $V$ such that there exist objects in $V$-categories with nontrivial automorphisms. I'd like to say this includes any cartesian enriching category which is not a poset, but I'm not quite sure.

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