# 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, 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.)