Very many mathematical notions (groups, rings, algebras, etc) form categories.

Similarly, a number of mathematical notions (spaces, spectra, derived schemes, etc.) form $(\infty,1)$-categories.
Their study is what modern algebraic topology is all about.

<i>To summarize:</i><br>
Modern algebraic topology is the study of everything that forms an $(\infty,1)$-category.