  The extension of ordinary category theory to $\infty$-categories at the start
of the 21st century was a spectacular achievement pioneered by Joyal and Lurie
with contributions from many others. Unfortunately, the technical arguments
required to solve the infinite homotopy coherence problems inherent in these
results make this theory difficult for non-experts to learn. This essay surveys
two programs that seek to narrow the gap between $\infty$-category theory and
ordinary 1-category theory. The first leverages similarities between the
categories in which 1-categories and $\infty$-categories live as objects to
provide "formal" proofs of standard categorical theorems. The second, which is
considerably more speculative, explores $\infty$-categories from new
"univalent" foundations closely related to homotopy type theory.
