Hi all! This is, as you can guess, a (very brief) post about the motivation for -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 to $latex 0
Definition 1: A simplicial set is an -category if for each , there is always a unique arrow that lifts a map along the inclusion .
Readers with some (informal) knowledge of -categories may note that an -category is an -category, with , such that all -morphisms are invertible for ; however, this definition seems to yield something different. One can interpret this definition as follows: for an -category , the -simplices of can be interpreted as morphisms. The problem is that there is no way to compose these maps. However, by the definition of an -category, a -simplex that links the composition to a map must exist. This says that is homotopic to ; hence using -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 -category are all -categories!
I plan on returning back to abstract algebra in the next post. See you then!