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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Simplicial homotopy theory is the study of homotopy theory by means of simplicial sets, but also the study of those properties of simplicial sets detectable by means of techniques adapted from topological homotopy theory.

simplex category, simplicial identities,

simplicial object,

nerve, singular simplicial complex

Kan fibration, Kan complex,

simplicial homotopy

simplicial homotopy group

geometric realization

geometric realization of categories

geometric realization of simplicial topological spaces

weak homotopy equivalence

model structure on simplicial sets

Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967)

Edward Curtis, Simplicial Homotopy Theory, Advances in Math., 6, (1971), 107 – 209.

Paul Goerss, Rick Jardine, Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1996)

Zhen Lin Low, Notes on homotopical algebra

Emily Riehl, Categorical Homotopy Theory