nLab homotopy theory and algebraic topology -- references

The following lists basic references on homotopy theory, algebraic topology and some ( , 1 ) (\infty,1) -category theory and homotopy type theory, but see these entries for more pointers.


Historical article at the origin of all these subjects:

On early developments from there, such as the eventual understanding of the notion of higher homotopy groups:

Topological homotopy theory

Textbook accounts of homotopy theory of topological spaces (i.e. via “point-set topology”):

Algebraic topology

On algebraic topology:


On constructive methods (constructive algebraic topology):

Lecture notes:

Survey of various subjects in algebraic topology:

Survey with relation to differential topology:

With focus on ordinary homology, ordinary cohomology and abelian sheaf cohomology:

Some interactive 3D demos:

Further pointers:

Abstract homotopy theory

On localization at weak equivalences to homotopy categories:

On localization via calculus of fractions:

On localization via model category-theory:

On localization (especially of categories of simplicial sheaves/simplicial presheaves) via categories of fibrant objects:

See also:

Lecture notes:

Introduction, from category theory to (mostly abstract, simplicial) homotopy theory:

See also:

Simplicial homotopy theory

On simplicial homotopy theory:

Basic (,1)(\infty,1)-category theory

On (∞,1)-category theory and (∞,1)-topos theory:

Basic homotopy type theory

On synthetic homotopy theory in homotopy type theory:


Textbook accounts:

For more see also at homotopy theory formalized in homotopy type theory.


Indications of open questions and possible future directions in algebraic topology and (stable) homotopy theory:

More regarding the sociology of the field (such as its folklore results):

Last revised on February 10, 2024 at 13:42:59. See the history of this page for a list of all contributions to it.