Motivation for Quasicategories

Hi all! This is, as you can guess, a (very brief) post about the motivation for \infty -categories.

Let us consider two (seemingly) different modifications of different objects. It is natural to ask if one can loosen the requirement that the extension in Proposition 3 of a previous post should be unique, to simply asking about existence. Also recall the definition of a Kan complex as given in Definition 5 of loc. cit. It is natural to ask to loosen the requirement that the lift exists for 0\leq i\leq n to $latex 0

Definition 1: A simplicial set X is an \infty -category if for each 0<i<n, there is always a unique arrow that lifts a map \Lambda^n_i\to K along the inclusion \Lambda^n_i\subset\Delta^n .

Readers with some (informal) knowledge of n -categories may note that an \infty -category is an n -category, with n=\infty, such that all k -morphisms are invertible for k>1; however, this definition seems to yield something different. One can interpret this definition as follows: for an \infty -category X, the 1 -simplices of X can be interpreted as morphisms. The problem is that there is no way to compose these maps. However, by the definition of an \infty -category, a 2 -simplex \Delta^2\to X that links the composition X\to Y\to Z to a map X\to Z must exist. This says that X\to Y\to Z is homotopic to X\to Z; hence using \infty -categories allows one to do things up to homotopy (for example, one can ask for homotopy commutativity of diagrams instead of strict commutativity). For those looking for examples: we have already provided two! Kan complexes and the nerve of a 1 -category are all \infty -categories!

I plan on returning back to abstract algebra in the next post. See you then!

S.D. (J.T.)

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