topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A simplicial topological group is a simplicial object in the category of topological groups.
For various applications the ambient category Top of topological spaces is taken specifically to be
the category of compactly generated weakly Hausdorff spaces, or
or the category of k-spaces.
We take Top to be the category of k-spaces in the following.
A simplicial topological group $G$ is called well-pointed if for $*$ the trivial simplicial topological group and $i : * \to G$ the unique homomorphism, all components $i_n : * \to G_n$ are closed cofibrations.
For $B \in Top$ a fixed base object, it is often desirable to work in “$B$-parameterized families”, hence in the over-category $Top/B$ (see MaySigurdson). There is the relative Strøm model structure on $Top/B$.
A simplicial group in $G$ in $Top/B$ is called well-sectioned if for $B$ the trivial simplicial topological group over $B$ and $i : B \to G$ the unique homomorphism, all components $i_n : B \to G_n$ are $\bar f$-cofibrations.
Recall for a discrete simplicial group $G$ the simplicial classifying space coprojection $W G \to \overline{W} G$, being a Kan complex presentation of the universal principal infinity-bundle $\mathbf{E}G \to \mathbf{B}G$ from simplicial group. These constructions for discrete simplicial groups have immediate analogs for simplicial topological groups.
Let $G$ be a simplicial topological group. Write $\bar W G \in Top^{\Delta^{op}}$ for the simplicial topological space whose topological space of $n$-simplices is the product
in Top, equipped with the evident face and degeneracy maps (see at simplicial classifying space).
We say a morphism $f : X \to Y$ of simplicial topological spaces is a global Kan fibration if for all $n \in \mathbb{N}$ and $0 \leq k \leq n$ the canonical morphism
$\Lambda^n_k \in$ sSet $\hookrightarrow Top^{\Delta^{op}}$ is the $k$th $n$-horn regarded as a discrete simplicial topological space:
$sTop(-,-) : sTop^{op} \times sTop \to Top$ is the Top-hom object.
We say a simplicial topological space $X_\bullet \in Top^{\Delta^{op}}$ is (global) Kan simplicial space if the unique morphism $X_\bullet \to *$ is a global Kan fibration, hence if for all $n \in \mathbb{N}$ and all $0 \leq i \leq n$ the canonical continuous function
into the topological space of $k$th $n$-horns admits a section.
This global notion of Kan simplicial spaces is considered for instance in (Brown & Szczarba 1989) and (May).
Let $G$ be a simplicial topological group. Then
$G$ is a globally Kan simplicial topological space;
$\bar W G$ is a globally Kan simplicial topological space;
$W G \to \bar W G$ is a global Kan fibration.
The first statement appears as (Brown & Szczarba 1989, theorem 3.8), the second is noted in (Roberts & Stevenson 2012), the third as (Brown & Szczarba 1989, lemma 6.7).
If $G$ is a well-pointed simplicial topological group (Def. ), then
$\overline{W} G$ is a proper simplicial topological space;
the geometric realization $|G|$ is well-pointed.
(Roberts & Stevenson 2012, Prop. 3, for the last item see also Baez & Stevenson 2008, Lem. 1)
Basic discussion of simplicial topological groups:
Edgar H. Brown, R. H. Szczarba?, Continuous cohomology and real homotopy type , Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 (doi:10.1090/S0002-9947-1989-0929667-6, pdf)
Peter May, Geometry of iterated loop spaces , SLNM 271, Springer-Verlag, 1972 (pdf)
Discussion of their geometric realization and principal $\infty$-bundles:
John Baez, Danny Stevenson, The Classifying Space of a Topological 2-Group, In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) Algebraic Topology_. Abel Symposia, vol 4. Springer 2009 (arXiv:0801.3843, doi:10.1007/978-3-642-01200-6_1)
David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460, nyjm:22-19)
Discussion of homotopy theory over a base $B$ is in
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