bundles

fiber bundle, fiber ∞-bundle

principal bundle, principal ∞-bundle

principal 2-bundle, principal 3-bundle

circle bundle, circle n-bundle

associated bundle, associated ∞-bundle

gerbe, 2-gerbe, ∞-gerbe,

local coefficient bundle

vector bundle, (∞,1)-vector bundle

universal principal ∞-bundle

subobject classifier, object classifier

bundle gerbe

groupal model for universal principal ∞-bundles

microbundle

Hopf fibration

prequantum circle bundle, prequantum circle n-bundle

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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

For $E_1, E_2 \to X$ two vector bundles, their direct sum over $X$, also called their Whitney sum, is the vector bundle $E_1 \oplus E_2 \to X$ whose fiber over any $x \in X$ is the direct sum of vector spaces of the fibers of $E_1$ and $E_2$.

Stiefel-Whitney class

virtual vector bundle

framed manifold, Thom spectrum