Around 1945, Leray introduced sheaves and spectral sequences. The relevant theory was further developed by Cartan, Serre, and others.
Eilenberg and MacLane introduced categories, functors, and natural transformations in 1945.
Ever since then, category theory played an increasingly important role in homotopy theory,
to the point where we are now often unable to cleanly separate them.
Around 1953, Cartan and Eilenberg completed their book on homological algebra (published in 1956).
Kan (advised by Eilenberg) systematically developed simplicial homotopy theory (and briefly also cubical homotopy theory)
starting from around 1955.
He introduced combinatorial homotopy groups, the Dold–Kan correspondence,
adjoint functors, limits and colimits, Kan extensions, etc.
In 1971, Gabriel and Ulmer published their systematic account of locally presentable categories.
Segal introduced Γ-spaces around 1972.
At the same time, May introduced operads, also in connection with infinite loop spaces.
Boardman and Vogt introduced quasicategories in 1973.
In 1977, Sullivan published his work on rational homotopy theory in the language of commutative differential graded algebras, complementing the previous work by Quillen.
Around 1979, Bousfield introduced what is now known as Bousfield localizations.
In 1983, Grothendieck introduced what is now known as Grothendieck homotopy theory, as well as derivators.
In 1989, Makkai and Paré published a systematic account of accessible categories.
In 1995, Baez and Dolan formulated the cobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture
about (∞,n)-categories for arbitrary n.
In the late 1990s, Voevodsky introduced and developed motivic homotopy theory (including some joint work with Morel).
Around the late 1990s, Smith introduced combinatorial model categories
and proved what is now known as the Smith recognition theorem
and established the existence of left Bousfield localizations of left proper combinatorial model categories.
Monoidal model categories were systematically studied by Schwede and Shipley starting from the late 1990s1997.
In the mid-2000s2006 (based on a 2003 preprint), Lurie's Higher Topos Theory came out, first as an online draft, which was later published.