Let $\mathrm{sSet}^+ = \mathrm{sSet}^+_{/ \Delta^0}$ be the model category of marked simplicial sets over the point. By Theorem 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 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?

  • $\begingroup$ 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. $\endgroup$ Sep 21, 2021 at 16:49
  • 1
    $\begingroup$ I realize that the forgetful functor, as a right Quillen functor, preserves fibrations. Why does it reflect fibrations though? $\endgroup$ Sep 21, 2021 at 17:12

1 Answer 1


Yes, this is true. There are various ways to prove this. Here's the shortest argument I can think of. One direction is easy to prove, so let's prove the other direction.

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.