In category theory, 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 model structure). [1] It is an ∞-category 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. [2]: 2–3 [3]
Alexander Grothendieck suggested in Pursuing Stacks [2]: 3–4, 201 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 by
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 ∞-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 . [6] There is an associated globular groupoid. Intuitively, the objects are the elements in , morphisms come from through the chain complex map , and higher -morphisms can be found from the higher chain complex maps . We can form a globular set with
One of the basic theorems about local systems is that they can be equivalently described as a functor from the fundamental groupoid to the category of abelian groups, the category of -modules, or some other abelian category. That is, a local system is equivalent to giving a functor
Another application of ∞-groupoids is giving constructions of n-gerbes and ∞-gerbes. Over a space an n-gerbe should be an object such that when restricted to a small enough subset , is represented by an n-groupoid, and on overlaps there is an agreement up to some weak equivalence. Assuming the homotopy hypothesis is correct, this is equivalent to constructing an object such that over any open subset
In category theory, 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 model structure). [1] It is an ∞-category 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. [2]: 2–3 [3]
Alexander Grothendieck suggested in Pursuing Stacks [2]: 3–4, 201 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 by
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 ∞-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 . [6] There is an associated globular groupoid. Intuitively, the objects are the elements in , morphisms come from through the chain complex map , and higher -morphisms can be found from the higher chain complex maps . We can form a globular set with
One of the basic theorems about local systems is that they can be equivalently described as a functor from the fundamental groupoid to the category of abelian groups, the category of -modules, or some other abelian category. That is, a local system is equivalent to giving a functor
Another application of ∞-groupoids is giving constructions of n-gerbes and ∞-gerbes. Over a space an n-gerbe should be an object such that when restricted to a small enough subset , is represented by an n-groupoid, and on overlaps there is an agreement up to some weak equivalence. Assuming the homotopy hypothesis is correct, this is equivalent to constructing an object such that over any open subset