Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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
Given a well-bahaved topological space $X$, its fundamental 2-groupoid is the 2-groupoid whose
1-morphisms are continuous paths $[0,1] \to X$ (or thin homotopy classes of such, to yield a strict 2-groupoid);
2-morphisms are homotopies between such paths, fixing their endpoints;
composition is given by concatenation of paths and homotopies.
This may equivalently be understood as:
the 2-groupoidal-categorification of the notion of the fundamental groupoid;
the homotopy 2-category of the fundamental $\infty$-groupoid;
At least if $X$ admits the structure of a CW-complex, its fundamental 2-groupoid captures the underlying 2-truncated weak homotopy type (the homotopy 2-type).
Original discussion of the fundamental 2-groupoid of Hausdorff topological spaces as a strict 2-groupoid (Grpd-enriched category):
and as a weak 2-groupoid:
Created on January 19, 2023 at 11:35:52. See the history of this page for a list of all contributions to it.