∞-groupoid

In , a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard ). It is an generalization of a groupoid, a category in which every morphism is an isomorphism. The homotopy hypothesis states that ∞-groupoids are equivalent to spaces up to homotopy. Alexander Grothendieck suggested in Pursuing Stacks that there should be an extraordinarily simple model of ∞-groupoids using globular sets, originally called hemispherical complexes. These sets are constructed as presheaves on the globular category . This is defined as the category whose objects are finite ordinals and morphisms are given bysuch that the globular relations holdThese encode the fact that -morphisms should not be able to see -morphisms. When writing these down as a globular set , the source and target maps are then written asWe can also consider globular objects in a category as functorsThere was hope originally that such a strict model would be sufficient for homotopy theory, but there is evidence suggesting otherwise. It turns out for its associated homotopy -type can never be modeled as a strict globular groupoid for . This is because strict ∞-groupoids only model spaces with a trivial Whitehead product. Given a topological space there should be an associated fundamental ∞-groupoid where the objects are points , 1-morphisms are represented as paths, 2-morphisms are homotopies of paths, 3-morphisms are homotopies of homotopies, and so on. From this infinity groupoid we can find an -groupoid called the fundamental -groupoid whose homotopy type is that of . Note that taking the fundamental ∞-groupoid of a space such that is equivalent to the fundamental n-groupoid . Such a space can be found using the Whitehead tower. One useful case of globular groupoids comes from a chain complex which is bounded above, hence let's consider a chain complex . There is an associated globular groupoid.