# Fibrations of fibrant marked simplicial sets

Let $$\mathrm{sSet}^+ = \mathrm{sSet}^+_{/ \Delta^0}$$ be the model category of marked simplicial sets over the point. By Theorem 3.1.5.1 in Higher Topos Theory, this model category is Quillen equivalent to $$\mathrm{sSet}$$ with Joyal's model structure. The fibrant objects of $$\mathrm{sSet}^+$$ are the quasicategories in which precisely the equivalences are marked.

My question concerns the classification of fibrations of fibrant objects in $$\mathrm{sSet}^+$$. For $$\mathrm{sSet}$$ with Joyal's model structure this is understood (Corollary 2.4.6.5 in Higher Topos Theory): A map $$f : X \rightarrow Y$$ of quasicategories $$X, Y$$ is a fibration if and only if $$f$$ is an inner fibration and an isofibration (equivalences in $$Y$$ can be lifted along a preimage of the codomain). Since lifts for equivalences are already part of this classification, I suspect that the following holds:

A map of fibrant marked simplicial sets is a fibration in $$\mathrm{sSet}^+$$ if and only if its underlying map of simplicial sets is a fibration in $$\mathrm{sSet}$$ with Joyal's model structure.

Is this true?

• That's true (better: the forgetful functor to sSet with the Joyal model structure is the right adjoint in a Quillen equivalence.) That's a very special case of HTT.3.1.5.1. Sep 21, 2021 at 16:49
• I realize that the forgetful functor, as a right Quillen functor, preserves fibrations. Why does it reflect fibrations though? Sep 21, 2021 at 17:12

Let $$U \colon \mathbf{sSet}^+ \to \mathbf{sSet}$$ denote the functor that forgets markings. We will use that the restriction of this functor to the full subcategory of fibrant objects in $$\mathbf{sSet}^+$$ (i.e. the naturally marked quasi-categories) is fully faithful and preserves cofibrations, fibrations, and weak equivalences. All of these properties are easy to prove.
Now, let $$f \colon A \to B$$ be a morphism of naturally marked quasi-categories, and suppose that $$U(f) \colon U(A) \to U(B)$$ is an isofibration of quasi-categories. We want to prove that $$f$$ is a fibration in the model structure on $$\mathbf{sSet}^+$$. Let $$f = p\circ j$$ be a factorisation of $$f$$ into a trivial cofibration $$j \colon A \to C$$ followed by a fibration $$p \colon C \to B$$ in $$\mathbf{sSet}^+$$. (Note that $$C$$ is fibrant, since $$B$$ is fibrant and $$p$$ is a fibration.) By the above preservation properties of $$U$$, $$U(f) = U(p) \circ U(j)$$ is a factorisation of $$U(f)$$ into a trivial cofibration followed by a fibration in the Joyal model structure on $$\mathbf{sSet}$$. But $$U(f)$$ is a fibration, so it is has the RLP wrt $$U(j)$$, and hence is a retract of $$U(p)$$ in $$\mathbf{sSet}$$. By the above fully faithfulness property of $$U$$ (restricted to the fibrant objects of $$\mathbf{sSet}^+$$), it follows that $$f$$ is a retract of $$p$$ in $$\mathbf{sSet}^+$$. Hence $$f$$ is a fibration in $$\mathbf{sSet}^+$$, since $$p$$ is.