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
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
A homotopy -type is a space where we consider its properties with regard to the fundamental groups of its components.
A continuous map is a homotopy -equivalence if it induces isomorphisms on and at each basepoint. Two spaces share the same homotopy -type if they are linked by a zig-zag chain of homotopy -equivalences.
For any space , you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space so that the inclusion of into is a homotopy -equivalence; up to (weak) homotopy equivalence, the result is the same for any space with the same homotopy -type. Accordingly, a homotopy -type may alternatively be defined as a space with trivial for , or as the unique (weak) homotopy type of such a space, or as its fundamental -groupoid (which will be a -groupoid).
See the general discussion in homotopy n-type.
A connected pointed homotopy 1-type is completely determined, up to (weak) homotopy equivalence, by the one group . A connected homotopy 1-type with is an Eilenberg-Mac Lane space and is often written . A general homotopy 1-type can then be written as a disjoint union of such s, and is completely determined by its fundamental groupoid.
In the other direction, for any (discrete) group one can construct its classifying space , which is a . In fact, many versions of this construction (such as the geometric realization of the simplicial nerve ; see Dold-Kan correspondence) apply just as well to any groupoid. We can obtain any 1-type in this way, since a groupoid is up to homotopy type (of groupoids!) a disjoint union of groups. However this description is not natural in the category of groupoids, and is analogous to choosing a basis for a vector space.
Moreover, every continuous map between s is induced by a group homomorphism, every map between 1-types is induced by a functor between groupoids, and every homotopy is induced by a conjugation (aka a natural transformation between groupoids). In fact, one can show that the -category of homotopy 1-types is equivalent to the 2-category Grpd of groupoids, via the above-described correspondence..
There are further aspects to this relationship; for instance, the van Kampen theorem for the fundamental groupoid shows how the algebra of groupoids models the gluing of spaces. The general result for non-connected spaces is possible because groupoids model homotopy 1-types, having structure in dimensions 0 and 1. For the search for algebraic structures that play an analogous role to groupoids for -types with , see the pages homotopy hypothesis, fundamental infinity-groupoid, cat-n-group, classifying space, crossed complex, and probably others.
homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|
h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |
h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |
h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |
h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |
h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |
h-level | -truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h--groupoid |
h-level | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h--groupoid |
Last revised on July 27, 2018 at 23:27:34. See the history of this page for a list of all contributions to it.