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What is the difference between:

($\infty,1$) categories - in which have for two objects you have an ($\infty,0$) category of morphisms (i.e. a space of morphisms)


categories weakly enriched over spaces - by that I mean categories such that hom(x,y) is always a space and composition is defined only up to (coherent) homotopy


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up vote 8 down vote accepted

If "space" means the same thing in the two cases, as seems to be implied, then there is no difference, at least not at that level of precision.

There are many different models of $(\infty,1)$-categories. Many of these, like $A_\infty$-categories, Segal categories, complete Segal spaces, and simplicial categories, try to make explicit the intuition that they are "categories weakly enriched over spaces," though in various different ways. Others, like quasicategories, are less directly related to the enrichment point of view, although they yield a provably equivalent theory (in a certain appropriate sense).

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In the quasi-category formalism of $(\infty,1)$-categories, it is hard to make sense of whether they are strictly enriched, or weakly enriched in infinity-groupoids. This is mostly due to the ambiguity (only up to homotopy) of what is meant by $Hom(X,Y)$. (There are various models for this, for example Hom^R(X,Y), Hom^L(X,Y), or the actual Hom in its associated simplicial category (apply the left-adjoint to the homotopy coherence nerve)). To avoid this, lets work with simplicial categories (which is justified as Bergner's model structure is Quillen equivalent to Joyal's). In this model for infinity-categories, the enrichment is STRICT. The fibrant cofibrant simplicial categories have each $Hom(X,Y)$ a Kan-complex = an infinity groupoid. The homotopy theory of simplicial categories models categories STRICTLY enriched in infinity-groupoids and WEAK functors between them. However, when people say Kan-complexes=infinity groupoids, they really mean they are WEAK infinity groupoids. (In a very concrete case, the tricategory of weak 2-groupoids is equivalent to the tricategory with CW-complexes as objects, continuous functions as morphisms, homotopies as 2-cells, and homotopy classes of homotopies between homotopies as 3-cells). Just as Mike said, the quasi-category approach (and hence the simplicial category approach) are provably equivalent to other approaches which take infinity-categories explicitly to be categories WEAKLY enriched in infinity-groupoids. Therefore, you can interpret this as a sort of strictification result. But, be careful, this is really a SEMI-strictification result; it is important that we are still enriching in WEAK infinity-groupoids. This could be thought of as a higher-dimensional analogue of the fact that, although not every tricategory is equivalent to a strict-3 category=category enriched in STRICT 2-categories, every tricategory is equivalent to a category enriched in weak 2-categories (to make this precise, every tricategory is equivalent to a category enriched in Gray see: However, in any rate, we still have to consider WEAK functors between such STRICTLY enriched guys. This is important to note (e.g. although every bicategory is equivalent to strict 2-category, there are more weak functors between them than strict ones). So the moral of the story is we can model $(\infty,1)$ categories as categories strictly enriched in weak infinitiy-groupoids with weak functors between them.

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