Simplicial objects and higher categories, part I

Hi all! Today I’ll introduce simplicial sets, and talk about how they relate to higher categories. Simplicial sets are basically combinatorial models for topological things; in fact, a particular kind of simplicial set (a Kan complex) is essentially equivalent to a topological space! So more general kinds of simplicial sets should model something more general than a topological space. Exploring this will be the topic of this post. The reader should note that these are directly from the (rather terse) notes I took while preparing for the Intel International Science and Engineering Fair last year (so this blog post is currently also serving as the notes for preparing for the Intel Science Talent Institute), so there may be small errors.

We will begin by talking about some combinatorial constructions. Let $latex [n]&s=1$ denote the set $latex {0,…,n}&s=1$ equipped with the linear ordering. Define $latex mathbf{Delta}&s=1$ to be the category whose objects are the sets $latex…