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) 


and


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


?



(∞,1) vs Category weakly enriched over spaces
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).


