I am going through V. Hinich's ["Lectures on Infinity Categories"](https://arxiv.org/pdf/1709.06271.pdf) and I have a (possibly trivial) question on Segal spaces.


We define Segal space to be a bisimplicial set $X$ which is fibrant in Reedy model structure and such that the "spine map" $X\_n\to X\_1\times\_{X\_0}X\_1\times\_{X\_0}\ldots \times\_{X\_0}X\_1$ is a weak equivalence of simplicial sets. Here by $X\_i$ we mean $X\_{i\ast}$.


Further on, when defining composition law in this model of infinity categories, we use the fact that
$$
X\_2\to X\_1\times\_{X\_0}X\_1
$$
is a trivial fibration.


The question is: that the last map is a weak equivalence is just the second axiom for Segal spaces. But how can I see that it is a fibration? I assume that it comes from the Reedy fibrancy assumption, but I fail to see how. Any help would be appreciated! If anybody knows a good reference, this will do the job too.



Reedy fibrancy and composition in Segal spaces
Given a simplicial object $X$ in a locally small category $\mathcal{M}$ and a simplicial set $S$, define the weighted limit $\{ S, X \}$ to be an object in $\mathcal{M}$ equipped with an isomorphism
$$\textrm{Hom}\_\mathcal{M} (-, \{ S, X \}) \cong \textrm{Hom}\_{\textbf{sSet}} (S, \textrm{Hom}\_\mathcal{M} (-, X))$$
of functors $\mathcal{M}^\textrm{op} \to \textbf{Set}$.


**Lemma.**
Let $X$ be a simplicial object in a complete model category $\mathcal{M}$.
The following are equivalent:


* $X$ is Reedy-fibrant.
* For every natural number $n$, the morphism $\{ \Delta^n, X \} \to \{ \partial \Delta^n, X \}$ induced by the inclusion $\partial \Delta^n \hookrightarrow \Delta^n$ is a fibration in $\mathcal{M}$.
* For every monomorphism $S \to T$ in $\textbf{sSet}$, the induced morphism $\{ T, X \} \to \{ S, X \}$ is a fibration in $\mathcal{M}$.


*Proof.*
When $\mathcal{M}$ has limits for small diagrams, $\{ S, X \}$ exists for all simplicial sets $S$, so for fixed $X$, $\{ {-}, X \} : \textbf{sSet}^\textrm{op} \to \mathcal{M}$ is right adjoint to $\textrm{Hom}\_\mathcal{M} (-, X) : \mathcal{M} \to \textbf{sSet}^\textrm{op}$.
In particular, $\{ {-}, X \} : \textbf{sSet}^\textrm{op} \to \mathcal{M}$ takes colimits in $\textbf{sSet}$ to limits in $\mathcal{M}$.
The second condition is now easily seen to be a paraphrase of the first condition, and the third condition is seen to be implied by the second condition by considering the skeletal filtration of a monomorphism.　◼


Thus, for example, if $X$ is a Reedy-fibrant bisimplicial set then the morphism you ask about is a Kan fibration because it is identifiable with morphism $\{ \Delta^2, X \} \to \{ \Lambda^2\_1, X \}$ induced by the inner horn inclusion $\Lambda^2\_1 \hookrightarrow \Delta^2$.
Similarly for the spine maps: they are the morphisms induced by the inclusion of the spine into $\Delta^n$.


