nLab H-spaceoid

Context

Homotopy theory

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

Higher algebra

Contents

Idea

The oidification of an H-space.

Definition

Classically

A H-spaceoid is a unital magmoid internal to the classical homotopy category of topological spaces Ho(Top), or in the homotopy category Ho(Top) *Ho(Top)_* of pointed topological spaces, which has a unit up to homotopy.

In homotopy type theory

In homotopy type theory, a H-spaceoid is a type AA with

  • For each a,b:Aa,b:A, a type hom A(a,b)hom_A(a,b), whose elements are called arrows or morphisms.

  • For each a:Aa:A, a morphism 1 a:hom A(a,a)1_a:hom_A(a,a), called the identity morphism.

  • For each a,b,c:Aa,b,c:A, a functor

    hom A(b,c)hom A(a,b)hom A(a,c)hom_A(b,c) \to hom_A(a,b) \to hom_A(a,c)

    called composition, and denoted infix by gfgfg \mapsto f \mapsto g \circ f, or sometimes gfgf.

  • For each a,b:Aa,b:A and f:hom A(a,b)f:hom_A(a,b), terms p:f=1 bfp:f = 1_b \circ f and q:f=f1 aq:f=f\circ 1_a.

See also

Last revised on June 7, 2022 at 17:40:55. See the history of this page for a list of all contributions to it.