homotopy theory

algebraic topology, simplicial homotopy theory

homotopy

homotopy type

stable homotopy theory

proper homotopy theory

directed homotopy theory

Pi-algebra, spherical object and Pi(A)-algebra

homotopy coherent category theory

homotopical category

model category

category of fibrant objects

Waldhausen category

homotopy category

(∞,1)-category

left homotopy

cylinder object

mapping cone

right homotopy

path object

mapping cocone

universal bundle

interval object

homotopy localization

infinitesimal interval object

homotopy group

fundamental group

Brown-Grossman homotopy group

categorical homotopy groups in an (∞,1)-topos

geometric homotopy groups in an (∞,1)-topos

fundamental ∞-groupoid

fundamental groupoid

fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos

fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos

fundamental (∞,1)-category

homotopy hypothesis-theorem

Hurewicz theorem

higher homotopy van Kampen theorem

Galois theory

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A homotopical functor $F : C \to D$ between two homotopical categories $C$ and $D$ is a functor which preserves the weak equivalences of $C$ (i.e., which sends all weak equivalences of $C$ to weak equivalences of $D$).

category with weak equivalences, homotopical category

simplicial localization