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What are the links between $\small{(\infty,1)}$-categories and simplicially enriched categories? For example I don't understand the "Idea" section on this nLab page

https://ncatlab.org/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category

When they say that $\mathscr C = LC$ I can only assume that there is a canonical "equivalence" between $\small (\infty,1)$-categories and simplicially enriched categories but I can't find any info on such a thing.

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The is a model structure on simplicial sets whose fibrant objects are "quasicategories", the simplicial sets in which every inner horn has a filler. If the 0-simplices are thought of as objects of a category and the 1-simplices as morphisms, then the horn filler condition gives a sense in which a quasicategory admits composites of all arrows, with composition associative up to a homotopy, which is itself well defined up to a higher homotopy, which is...and so on. The latter is the primeval concept of an $(\infty,1)$-category, and quasicategories are the most used model.

In fact the first model category for $(\infty,1)$-categories that was introduced was that of Bergner, which is on the category of simplicially enriched categories. This models an $(\infty,1)$-category as a "category" with morphisms of all dimensions, all admitting strictly associative compositions. It is very far from obvious that everything we'd like to call an $(\infty,1)$-category is equivalent to something of this form, but in fact this is the case: there is a Quillen equivalence between the model categories of simplicial categories and of quasicategories, showing that every $(\infty,1)$-category (viewed as a quasicategory) can be viewed as a simplicially enriched category.

The sense in which a quasicategory $Q$ is "equivalent" to some simplically enriched category $\mathcal C$ is that the homotopy categories $\mathrm{Ho}(Q)$ and $\mathrm{Ho}(\mathcal C)$ are equivalent, but also, roughly speaking, that $Q$ and $\mathcal C$ have the same mapping spaces. One can't formalize this directly by asking that there be a map inducing such equivalences $Q\to \mathcal C$, as they don't live in the same category, which is the reason for the introduction of the Quillen equivalence between the model structures.

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Not really a full answer, but it was too long to fit in the comments:

Firstly everything that you may want to know about the connection between various ides for $(\infty,1)$-categories is done by Lurie in Higher Topos Theory (mostly chapter 1).

Loosely what happens is that $(\infty,1)$-categories should be categories "weakly enriched" in $\infty$-groupoids. By weakly enriched, I mean that the hom-sets should be $\infty$-groupoids, but the induced composition need not be associative or unital on the nose, but only up to equivalence in those. There to my knowledge no way to do make this weak enrichment stuff very precise in full generality, but in the context of $(\infty,1)$-categories there are various ways of dealing with it.

Under the homotopy hypothesis, the $\infty$-groupoids are equivalent to homotopy types so an idea would be to try and enrich over homotopy types. Unfortunately, there is very little we can do with homotopy types in the abstract. However what we know very well is topological spaces, so a nice workaround is to use categories enriched in topological sets. In order to recover a weak enrichment over homotopy type, you can use the model category machinerie: under the right assumptions, when you localise an enriched category, the homotopy category you get is enriched over the localisation of the original enrichment. There are at least 2 ways (presented by Lurie) to achieve that : topological categories, which are categories enriched over topological sets, and simplicially enriched categories, which are categories enriched over simplicial sets. The reason one would consider simplicial here is, as far as I understand, because they carry a model structure whose homotopy category is the homotopy type (this is a theorem due to Quillen), and they make things manageable by combinatorial arguments.

Another approach find its origin again in Quillen's work (I think). For that recall that homotopy types can be described directly inside simplicial sets as being the Kan complexes, so in fact one can encode $\infty$-groupoids as the Kan complexes. Moreover, there is a construction encoding categories inside of simplicial sets, and since $(\infty,1)$-categories, should really be "$\infty$-groupoids whose first level is a category", it makes sense to combine these two conditions, and that is how you get the $(\infty,1)$-categories that Lurie defines (aka quasicategories).

With all this in mind, simplicial categories should be equivalent to $(\infty,1)$-categories, at least when taking homotopy categories, I expect the connection to go along the line of the proof that the homotopy category of simplicial is equivalent to Kan complexes, but "lifted of one dimension". Again all these are made precise in Lurie's book, and my understanding is a bit partial, but I think the intuition is correct

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