There are two ways to think about (∞, 1)-categories. The first is that an (∞, 1)- category, as its name suggests, should be some kind of higher categorical structure. The second is that an (∞, 1)-category should encode the data of a homotopy theory. So we first need to know what a homotopy theory is, and what a higher category is.

We can begin with the classical homotopy theory of topological spaces. In this setting, we consider topological spaces up to homotopy equivalence, or up to weak homotopy equivalence. Techniques were developed for defining a nice homotopy category of spaces, in which we define morphisms between spaces to be homotopy classes of maps between CW complex replacements of the original spaces being considered. However, the general framework here is not unique to topology; an analogous situation can be found in homological algebra. We can take projective replacements of chain complexes, then chain homotopy classes of maps, to define the derived category, the algebraic analogue of the homotopy category of spaces.

The question of when we can make this kind of construction (replacing by some particularly nice kinds of objects and then taking homotopy classes of maps) led to the definition of a model category by Quillen in the 1960s [100]. The essential information consists of some category of mathematical objects, together with some choice of which maps are to be designated as weak equivalences; these are the maps we would like to think of as invertible but may not be. The additional data of a model structure, and the axioms this data must satisfy, guarantee the existence of a well-behaved homotopy category as we have in the above examples, with no set-theoretic problems arising.

A more general notion of homotopy theory was developed by Dwyer and Kan in the 1980s. Their simplicial localization [57] and hammock localization [56] constructions provided a method in which a category with weak equivalences can be assigned to a simplicial category, or category enriched in simplicial sets. More remarkably, they showed that up to a natural notion of equivalence (now called Dwyer–Kan equivalence), every simplicial category arises in this way [55]. Thus, if a “homotopy theory” is just a category with weak equivalences, then we can think of simplicial categories as homotopy theories. In other words, simplicial categories provide a model for homotopy theories.