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It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a recent result of Lennart Meier, a certain "double …

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 $ …

I assume $\varinjlim_{j : \mathcal{J}} \mathcal{I}_j = \mathcal{I}$ is meant in the strict sense of 1-categories. Since $\textbf{Cat}$ is cartesian closed, $$\textstyle [\mathcal{I}, \mathcal{C}] \con …

We can detect the difference between the two constructions using the homotopy category. Given any simplicially enriched category $\mathcal{C}$, we can construct an ordinary category $\pi_0 [\mathcal{C …

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent …