algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

, the fundamental

groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial func ...

is a certain

topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

of a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the

homotopy type In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

of a topological space. In terms of category theory, the fundamental groupoid is a certain

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...

from the category of topological spaces to the category of

Definition

Let be a

. Consider the equivalence relation on continuous paths in in which two continuous paths are equivalent if they are

homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

with fixed endpoints. The fundamental groupoid assigns to each ordered pair of points in the collection of equivalence classes of continuous paths from to . More generally, the fundamental groupoid of on a set restricts the fundamental groupoid to the points which lie in both and . This allows for a generalisation of the

Van Kampen theorem A van is a type of road vehicle used for transporting goods or people. Depending on the type of van, it can be bigger or smaller than a pickup truck and SUV, and bigger than a common car. There is some varying in the scope of the word across ...

using two base points to compute the fundamental group of the circle. As suggested by its name, the fundamental groupoid of naturally has the structure of a

. In particular, it forms a category; the objects are taken to be the points of and the collection of morphisms from to is the collection of equivalence classes given above. The fact that this satisfies the definition of a category amounts to the standard fact that the equivalence class of the concatenation of two paths only depends on the equivalence classes of the individual paths. Likewise, the fact that this category is a groupoid, which asserts that every morphism is invertible, amounts to the standard fact that one can reverse the orientation of a path, and the equivalence class of the resulting concatenation contains the constant path. Note that the fundamental groupoid assigns, to the ordered pair , the fundamental group of based at .

Basic properties

Given a topological space , the path-connected components of are naturally encoded in its fundamental groupoid; the observation is that and are in the same path-connected component of if and only if the collection of equivalence classes of continuous paths from to is nonempty. In categorical terms, the assertion is that the objects and are in the same groupoid component if and only if the set of morphisms from to is nonempty. Suppose that is path-connected, and fix an element of . One can view the fundamental group as a category; there is one object and the morphisms from it to itself are the elements of . The selection, for each in , of a continuous path from to , allows one to use concatenation to view any path in as a loop based at . This defines an

equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences f ...

between and the fundamental groupoid of . More precisely, this exhibits as a skeleton of the fundamental groupoid of . The fundamental groupoid of a (path-connected) differentiable manifold is actually a

Lie groupoid In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smo ...

, arising as the gauge groupoid of the

universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...

of .

Bundles of groups and local systems

Given a topological space , a ''

local system In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology ...

'' is a

from the fundamental groupoid of to a category. As an important special case, a ''bundle of (abelian) groups'' on is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on assigns a group to each element of , and assigns a

group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...

to each continuous path from to . In order to be a functor, these group homomorphisms are required to be compatible with the topological structure, so that homotopic paths with fixed endpoints define the same homomorphism; furthermore the group homomorphisms must compose in accordance with the concatenation and inversion of paths. One can define homology with coefficients in a bundle of abelian groups.Whitehead, section 6.2. When satisfies certain conditions, a local system can be equivalently described as a locally constant sheaf.

Examples

* The fundamental groupoid of the

singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...

space is the trivial groupoid (a groupoid with one object * and one morphism * The fundamental groupoid of the

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

is connected and all of its vertex groups are isomorphic to

(\mathbb,+)

, the

additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structures ...

integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

The homotopy hypothesis

The

homotopy hypothesis In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geometric realizations of these sets give mod ...

, a well-known conjecture in homotopy theory formulated by Alexander Grothendieck, states that a suitable

generalization A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characte ...

of the fundamental groupoid, known as the fundamental

∞-groupoid 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). I ...

, captures ''all'' information about a topological space up to

weak homotopy equivalence In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category with cla ...

References

* Ronald Brown
Topology and groupoids.
Third edition of ''Elements of modern topology'' cGraw-Hill, New York, 1968 With 1 CD-ROM (Windows, Macintosh and UNIX). BookSurge, LLC, Charleston, SC, 2006. xxvi+512 pp. * Brown, R., Higgins, P. J. and Sivera, R., Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids.'' Tracts in Mathematics Vol 15. European Mathematical Society (2011). (663+xxv pages) '' * J. Peter May
A concise course in algebraic topology.
Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999. x+243 pp. * Edwin H. Spanier. Algebraic topology. Corrected reprint of the 1966 original. Springer-Verlag, New York-Berlin, 1981. xvi+528 pp. * George W. Whitehead. Elements of homotopy theory. Graduate Texts in Mathematics, 61. Springer-Verlag, New York-Berlin, 1978. xxi+744 pp.

External links

* The website of Ronald Brown, a prominent author on the subject of groupoids in topology: http://groupoids.org.uk/ * * Higher category theory Algebraic topology {{topology-stub