nLab
algebraic topology

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Topology

topology

algebraic topology

Basic concepts

Theorems

Extra stuff, structure, properties

Examples

Contents

Idea

Algebraic topology is generally the study of functors from nice categories of spaces to algebraic categories. This is in hindsight, in fact, category theory originally developed out of algebraic topology, where it was used first simply to describe what was going on and then to axiomatise Eilenberg-Steenrod cohomology theories.

References

A textbook with an emphasis on homotopy theory is in

  • Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2002)

Lecture notes include

Online resources include

Revised on October 17, 2012 23:33:05 by Urs Schreiber (82.169.65.155)
Edit | Back in time (7 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: Timeline of category theory and related mathematics, oriental, foundation of mathematics, homotopy theory, mathematics, anafunctor, groupoid cardinality, Top, infinity-groupoid, model category, simplicial set, simplicial category, category with weak equivalences, homotopy category, derived functor, cubical set, Nonabelian Algebraic Topology, (infinity,1)-category, operad, Kan complex, n-groupoid, simplicial object, quasi-isomorphism, topological space, nice topological space, CW complex, homotopical category, category of fibrant objects, deformation retract, homotopical functor, model structure on topological spaces, simplex, topology, simplicially enriched category, path space object, horn, geometric realization, simplicial complex, locale, sober topological space, interval object, fundamental category, directed topological space, Kan fibration, Hopf algebra, simplicial T-complex, small object argument, homotopy hypothesis, fibration, (infinity,n)-category, simplicial group, simplicial identities, fundamental infinity-groupoid, directed homotopy theory, homotopy 3-type, homotopy n-type, simplicial presheaf, homotopy limit, Grpd, SimpSet, van Kampen theorem, filtered topological space, connected filtered space, higher van Kampen theorem, homotopy pullback, homotopy 2-type, homotopy type, Euler characteristic, 1-groupoid, join of simplicial sets, T-complex, cubical T-complex, topological group, homotopy group, fundamental group, m-cofibrant space, sequential topological space, compactly generated topological space, Postnikov system, homotopy equivalence, weak homotopy equivalence, convenient category of topological spaces, Sierpinski space, specialization topology, Waldhausen category, fundamental group of a topos, shape theory, contractible space, etale space, Thomason model structure, homotopy category of an (infinity,1)-category, derived category, mapping cone, category of open subsets, cohomology, Reedy category, convergence space, Cauchy space, reduced suspension, generalized Reedy category, differential forms on simplices, calculus of fractions, associahedron, category of simplices, Eckmann-Hilton duality, uniform space, compact space, topological stack, Hausdorff space, Freudenthal suspension theorem, compact Lie group, connected space, local homeomorphism, locally compact space, neighborhood, simplicial homotopy group, boundary of a simplex, topological vector space, Eilenberg-Mac Lane space, tmf, spectral sequence, loop space, Carlos Simpson, Galois theory, numerable open cover, generalized (Eilenberg-Steenrod) cohomology, algebraic topology, weak inverse, simplicial skeleton, homotopy, homotopy (as an operation), model structure on presheaves of simplicial groupoids, Vladimir Voevodsky, H-space, suspension, nerve and realization, singular simplicial complex, sphere, paracompact topological space, second-countable space, Kan fibrant replacement, (infinity,1)-categorical hom-space, symmetric function, general topology, compactum, anodyne morphism, Andrew Ranicki, Hurewicz connection, continuous map, singular cohomology, cartesian space, proximity space, formal group, stable homotopy theory, derivator, homotopy - contents, compact-open topology, cochain on a simplicial set, free groupoid, n-truncated object of an (infinity,1)-category, Dold fibration, A-infinity-space, neighborhood retract, A1-homotopy theory, separable space, subspace topology, homotopy localization, spine, Whitehead tower, string group, dendroidal set, knot, Whitehead theorem, Hurewicz fibration, quasifibration, suspension object, cocylinder, Hurewicz cofibration, mapping cylinder, exponential law for spaces, homotopy extension property, one-point compactification, homotopy inverse, Serre fibration, basic problems of algebraic topology, mapping cocone, delayed homotopy, internal infinity-groupoid, foliation, linking number, simplicial ring, isotopy, cofibration, open cover, Allen Hatcher, Euclidean-topological infinity-groupoid, sequentially compact space, topology - contents, automorphism infinity-group, stacked cover, open subspace, Stone space, categorical homotopy groups in an (infinity,1)-topos, simplicial topological group, quotient space, partition of unity, Paul Bressler, Serre-Swan theorem, CW approximation, test category, model structure on cubical sets, simplicial localization of a homotopical category, open map, Loc, Calculus of fractions and homotopy theory, locally contractible space, decalage, clique, Ho(Top), Warsaw circle, proper homotopy theory, coherent topological space, homotopy Kan extension, closed subspace, string topology, Thom space, Jerome Kaminker, slant product, transgression, pointed derivator, zigzag, good open cover, subdivision, Cantor space, locally finite cover, nerve theorem, two-out-of-six property, topological base, homotopy exact square, pullback in a derivator, nice simplicial topological space, geometric homotopy groups in an (infinity,1)-topos, topological locale, frame of opens, G-delta subspace, infinite loop space, dense subspace, fundamental theorem of algebra, Teimuraz Pirashvili, formal topology, homotopy equivalence of toposes, n-connected space, long exact sequence of homotopy groups, weak topology, shrinkable map, Andrey Lazarev, factorization lemma, discrete and codiscrete topology, geometric realization of simplicial topological spaces, simplicial topological space, augmented simplicial set, Arne Ostvaer, cosimplicial simplicial set, proper Lie groupoid, ball, fundamental (infinity,1)-category, math resources by individuals, strict 2-groupoid, zigzag category, geometric realization of cohesive infinity-groupoids, boundary, topological groupoid, torus, link, chain homotopy, Reidemeister move, circle, TopMfd, model structure on simplicial algebras, interior, convex set, finite topological space, Haefliger groupoid, holonomy groupoid, covering dimension, Euclidean topology, higher homotopy van Kampen theorem, polyhedron, cell complex, model structure on operator algebras, Vassiliev skein relation, van Kampen theorem for toposes, plane, polytope, annulus, fundamental infinity-groupoid in a locally infinity-connected (infinity,1)-topos, Pi-algebra, spherical object, clopen set, Peter Hilton, Boardman-Vogt resolution, Kontsevich integral, free loop space, singular homology, localizing subcategory, proper topological groupoid, L.E.J. Brouwer, geometric realization of categories, Myles Tierney, Milnor mu-bar invariants, Betti number, writhe, Heyting dimension, homotopy groups of a Lie groupoid, cohomology operation, locally equi-connected space, A Concise Course in Algebraic Topology, Differential Forms in Algebraic Topology, locally contractible topological infinity-groupoid, two-out-of-three, Homotopy Limit Functors on Model Categories and Homotopical Categories, semi-simplicial set, operator topology, relational beta-module, homotopy type theory, Parametrized Homotopy Theory, singular knot, Manifold Atlas, Goldman bracket, C-infinity topology, looping, locally compact topological group, closed map, Tychonoff space, dendroidal homotopy coherent nerve, product topological space, map from compact space to Hausdorff space, bisimplicial set, contents of contents, shift map, B1-homotopy theory, homotopy groups of spheres, suspension of a chain complex, pointless topology, Hurewicz theorem, homotopy level, maximal compact subgroup, function extensionality, higher inductive type, coend in a derivator, focal point, univalence axiom, Delta-generated space, Bousfield lattice, Hopf fibration, Sierpinski topos, Cartan-Eilenberg category, almost connected topological group, dense subalgebra, elegant Reedy category, quadratic form, compact topological group, Simplicial homotopy theory, simplicial homotopy theory, iterated loop space, schedule, Eilenberg subcomplex, model structure for infinity-groupoids, closed cover, local fibration, reduced simplicial set, Heine-Borel theorem, strongly compact topological space, projective resolution, sequence space, null homotopy, first-countable space, interval object in chain complexes, syntopogenous space, low-dimensional topology, homotopy class, homotopy category of chain complexes, Ho(sSet), pseudocircle, free resolution, n-connected map, homotopical structure on C*-algebras, topological ring, limit point, h-groupoid, model topos, practical foundation of mathematics, degree of a continuous function, localization of model categories, model structure for homotopy n-types, Toda bracket, differential Galois theory, semi-simplicial object, core in a 2-category, singular foliation, Hopf construction, topological algebra, convergence, noncommutative topology, relative category, convergence vector space, Kan-Thurston theorem