  Naturally occurring diagrams in algebraic topology are commutative up to
homotopy, but not on the nose. It was quickly realized that very little can be
done with this information. Homotopy coherent category theory arose out of a
desire to catalog the higher homotopical information required to restore
constructibility (or more precisely, functoriality) in such "up to homotopy"
settings. These notes provide a three-part introduction to homotopy coherent
category theory. The first part surveys the classical theory of homotopy
coherent diagrams of topological spaces. The second part introduces the
homotopy coherent nerve and connects it to the free resolutions used to define
homotopy coherent diagrams. This connection explains why diagrams valued in
homotopy coherent nerves or more general $\infty$-categories are automatically
homotopy coherent. The final part ventures into homotopy coherent algebra,
connecting the newly discovered notion of homotopy coherent adjunction to the
classical cobar and bar resolutions for homotopy coherent algebras.
