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Suppose $C$ is a Segal category, that is a functor $X: \Delta^{op} \rightarrow Spaces$ so that $X_0$ is discrete and satisfies the Segal condition $X_k \cong X_1 \times_{X_0} \cdots \times_{X_0} X_1$. This is a model for an $\infty$-category. To compare it to other models of $\infty$-categories, like quasi-categories, one can take the first row, i.e. the set of points $X_{k,0}$ of each space $X_k$, and the result will be a simplicial set (this is proven in Joyal-Tierney, Quasi-categories vs. Segal spaces https://arxiv.org/abs/math/0607820).

However, to make this work, my understanding is that one first needs to take a Reedy fibrant replacement of $X$ in the Reedy model category of simplicial spaces, and then take the first row.

Question Is there an 'explicit' way to take a Reedy fibrant replacement of a Segal category? Note that this will not change the homotopy type of each space $X_k$.

Any explanation of how this Reedy fibrant replacement procedure actually works would be helpful. Otherwise, I am not sure how to compare my Segal category to a quasi-category.

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Question Is there an 'explicit' way to take a Reedy fibrant replacement of a Segal category? Note that this will not change the homotopy type of each space Xk.

The explicit fibrant replacement functor is already present on the original paper of Reedy on Reedy model structures. It is constructed inductively on the degree of indexing objects, and the very point of Reedy model structures is that the special properties of Reedy categories allow for such an inductive construction.

The article Reedy model structure has references to the original article and further expositions.

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