nLab balanced product

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Contents

Context

Category theory

category theory

Concepts

category
functor
natural transformation
Cat

Universal constructions

universal construction

Theorems

Yoneda lemma
Isbell duality
Grothendieck construction
adjoint functor theorem
monadicity theorem
adjoint lifting theorem
Tannaka duality
Gabriel-Ulmer duality
small object argument
Freyd-Mitchell embedding theorem
relation between type theory and category theory

Extensions

sheaf and topos theory
enriched category theory
higher category theory

Applications

applications of (higher) category theory

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Topology

topology (point-set topology, point-free topology)

Introduction

Basic concepts

open subset, closed subset, neighbourhood
topological space, locale
base for the topology, neighbourhood base
finer/coarser topology
closure, interior, boundary
separation, sobriety
continuous function, homeomorphism
uniformly continuous function
embedding
open map, closed map
sequence, net, sub-net, filter
convergence
category Top
- convenient category of topological spaces

Universal constructions

initial topology, final topology
subspace, quotient space,
fiber space, space attachment
product space, disjoint union space
mapping cylinder, mapping cocylinder
mapping cone, mapping cocone
mapping telescope
colimits of normal spaces

Extra stuff, structure, properties

nice topological space
metric space, metric topology, metrisable space
Kolmogorov space, Hausdorff space, regular space, normal space
sober space
compact space, proper map

sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
compactly generated space
second-countable space, first-countable space
contractible space, locally contractible space
connected space, locally connected space
simply-connected space, locally simply-connected space
cell complex, CW-complex
pointed space
topological vector space, Banach space, Hilbert space
topological group
topological vector bundle, topological K-theory
topological manifold

Examples

empty space, point space
discrete space, codiscrete space
Sierpinski space
order topology, specialization topology, Scott topology
Euclidean space
- real line, plane
cylinder, cone
sphere, ball
circle, torus, annulus, Moebius strip
polytope, polyhedron
projective space (real, complex)
classifying space
configuration space
path, loop
mapping spaces: compact-open topology, topology of uniform convergence
- loop space, path space
Zariski topology
Cantor space, Mandelbrot space
Peano curve
line with two origins, long line, Sorgenfrey line
K-topology, Dowker space
Warsaw circle, Hawaiian earring space

Basic statements

Hausdorff spaces are sober
schemes are sober
continuous images of compact spaces are compact
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces are normal
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
closed injections are embeddings
proper maps to locally compact spaces are closed
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and sigma-compact spaces are paracompact
locally compact and second-countable spaces are sigma-compact
second-countable regular spaces are paracompact
CW-complexes are paracompact Hausdorff spaces

Theorems

Urysohn's lemma
Tietze extension theorem
Tychonoff theorem
tube lemma
Michael's theorem
Brouwer's fixed point theorem
topological invariance of dimension
Jordan curve theorem

Analysis Theorems

Heine-Borel theorem
intermediate value theorem
extreme value theorem

topological homotopy theory

left homotopy, right homotopy
homotopy equivalence, deformation retract
fundamental group, covering space
fundamental theorem of covering spaces
homotopy group
weak homotopy equivalence
Whitehead's theorem
Freudenthal suspension theorem
nerve theorem
homotopy extension property, Hurewicz cofibration
cofiber sequence
Strøm model category
classical model structure on topological spaces

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

Idea

The balanced product is the natural tensor product in the category of topological G-spaces (for $G$ any topological group) with values in plain topological spaces.

Similarly there is a balanced product of smooth G-manifolds for $G$ any Lie group, etc.

Examples

Example

The balanced product of a $G$ -principal bundle with a G-space $F$ is the associated $F$ -fiber bundle.

Related concepts

balanced smash product
associated bundle

References

Stephen Mitchell, Notes on principal bundles and classifying spaces, Lecture Notes. University of Washington, 2011 (pdf, pdf)

Last revised on November 27, 2025 at 06:13:40. See the history of this page for a list of all contributions to it.

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