Cisinski trivial fibrations and the subobject classifier

Recall that a Cisinski model structure is a cofibrantly generated model structure on a Grothendieck topos whose cofibrations are the monomorphisms. Accordingly, a Cisinski trivial fibration is a morphism that has the right lifting property with respect to all monomorphisms; note that this makes sense in any category, but we will mainly focus on the case of elementary toposes.

Also recall that a subobject classifier in a category with pullbacks is an object Ω equipped with a natural bijection between morphisms XΩ and isomorphism classes of subobjects of X. It is easy to see that any subobject classifier Ω is an injective object with respect to the class of monomorphisms, and so in a category with finite limits, the projection Ω × XX is a Cisinski trivial fibration (by the stability of right lifting properties under pullback). If, in addition, the category has a strict initial object, then Ω has two disjoint global points and is therefore an interval object. Cisinski calls this the Lawvere cylinder.

Putting these observations together, we may deduce the following:

Proposition. For an elementary topos, inverting Cisinski trivial fibrations is the same thing as quotienting out the congruence of homotopy with respect to the Lawvere cylinder.

In particular, every homotopy equivalence with respect to the Lawvere cylinder is a weak equivalence in any Cisinski model structure. Joyal’s determination principle for model structures implies that a Cisinski model structure is entirely determined by its fibrant objects, and the above proposition implies that the homotopy category of a Cisinski model category is equivalent to a reflective subcategory of the homotopy category with respect to the Lawvere cylinder, where the reflector is induced by fibrant replacement.

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