“Groupoid”的意思、由来-开放百科全书

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:

In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g : A~>B, h : B~>C, say. Composition is then a total function: o : A~>B -> B~>C -> A~>C, so that g o h : A~>C.

Groupoids are often used to reason about geometrical objects such as manifolds. {{harvs|txt|first=Heinrich |last=Brandt|authorlink=Heinrich Brandt|year=1927}} introduced groupoids implicitly via Brandt semigroups.^[2]

Definitions

A groupoid is an algebraic structure

consisting of a non-empty set

and a binary partial function '

' defined on

Algebraic

A groupoid is a set

with a unary operation

and a partial function

. Here * is not a binary operation because it is not necessarily defined for all possible pairs of

-elements. The precise conditions under which

is defined are not articulated here and vary by situation.

and ⁻¹ have the following axiomatic properties. Let

, and

be elements of

. Then:

Category theoretic

A groupoid is a small category in which every morphism is an isomorphism, i.e. invertible.^[1] More precisely, a groupoid G is:

If f is an element of G(x,y) then x is called the source of f, written s(f), and y the target of f (written t(f)).

More generally, one can consider a groupoid object in an arbitrary category admitting finite fiber products.

Comparing the definitions

The algebraic and category-theoretic definitions are equivalent, as follows. Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y).

Then

and

become partially defined operations on G, and

will in fact be defined everywhere; so we define ∗ to be

and ⁻¹ to be

. Thus we have a groupoid in the algebraic sense. Explicit reference to G₀ (and hence to

) can be dropped.

Conversely, given a groupoid G in the algebraic sense, and define an equivalence relation

between its elements as

Let G₀ be the set of equivalence classes of

, that is,

. Denote a ∗ a⁻¹ with

with

The identity morphism on x is then

, and the category-theoretic inverse of f is f⁻¹.

Sets in the definitions above may be replaced with classes, as is generally the case in category theory.

Vertex groups

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G(x,x), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Category of groupoids

A subgroupoid is a subcategory that is itself a groupoid. A groupoid morphism is simply a functor between two (category-theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd.

It is useful that this category is, like the category of small categories, Cartesian closed. That is, we can construct for any groupoids

a groupoid C

whose objects are the morphisms

and whose arrows are the natural equivalences of morphisms. Thus if

are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids

there is a natural bijection

Fibrations and coverings

Particular kinds of morphisms of groupoids are of interest. A morphism

of groupoids is called a fibration if for each object

and each morphism

starting at

there is a morphism

starting at

such that

. A fibration is called a covering morphism or covering of groupoids if further such an

is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering maps of spaces.^[4]

It is also true that the category of covering morphisms of a given groupoid

is equivalent to the category of actions of the groupoid

on sets.

Examples

Linear algebra

Given a field K, the corresponding general linear groupoid GL_*(K) consists of all invertible matrices, of any size, whose entries range over K. Matrix multiplication interprets composition. If G = GL_*(K), then the set of natural numbers is a proper subset of G₀, since for each natural number n, there is a corresponding identity matrix of dimension n. G(m,n) is empty unless m=n, in which case it is the set of all nxn invertible matrices.

Topology

Given a topological space

, let

be the set

. The morphisms from the point

to the point

are equivalence classes of continuous paths from

, with two paths being equivalent if they are homotopic.

Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of

, denoted

(or sometimes,

).^[5] The usual fundamental group

is then the vertex group for the point

An important extension of this idea is to consider the fundamental groupoid

where

is a chosen set of "base points". Here, one considers only paths whose endpoints belong to

is a sub-groupoid of

. The set

may be chosen according to the geometry of the situation at hand.

Equivalence relation

is a set with an equivalence relation denoted by infix

, then a groupoid "representing" this equivalence relation can be formed as follows:

Group action

If the group

acts on the set

, then we can form the action groupoid (or transformation groupoid) representing this group action as follows:

More explicitly, the action groupoid is a small category with

and

with source and target maps

and

. It is often denoted

(or

). Multiplication (or composition) in the groupoid is then

which is defined provided

For

, the vertex group consists of those

with

, which is just the isotropy subgroup at

for the given action (which is why vertex groups are also called isotropy groups).

Another way to describe

-sets is the functor category

, where

is the groupoid (category) with one element and isomorphic to the group

. Indeed, every functor

of this category defines a set

and for every

(i.e. for every morphism in

) induces a bijection

. The categorical structure of the functor

assures us that

defines a

-action on the set

. The (unique) representable functor

→

is the Cayley representation of

. In fact, this functor is isomorphic to

and so sends

to the set

which is by definition the "set"

and the morphism

(i.e. the element

) to the permutation

of the set

. We deduce from the Yoneda embedding that the group

is isomorphic to the group

, a subgroup of the group of permutations of

Fifteen puzzle

The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed).^[6]^[7]^[8] This groupoid acts on configurations.

Mathieu groupoid

The Mathieu groupoid is a groupoid introduced by John Horton Conway acting on 13 points such that the elements fixing a point form a copy of the Mathieu group M₁₂.

Relation to groups

If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group.^[9] Many concepts of group theory generalize to groupoids, with the notion of functor replacing that of group homomorphism.

is an object of the groupoid

, then the set of all morphisms from

forms a group

(called the vertex group, defined above). If there is a morphism

from

, then the groups

and

are isomorphic, with an isomorphism given by the mapping

Every connected groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to an action groupoid (as defined above) (

) [by connectedness, there will only be one orbit under the action]. If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups

and sets

for each connected component).

Note that the isomorphism described above is not unique, and there is no natural choice. Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object

, a group isomorphism

from

, and for each

other than

, a morphism in

from

In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a multiset of unrelated groups. In other words, for equivalence instead of isomorphism, one need not specify the sets

, only the groups

Consider the examples in the previous section. The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various general linear groups

. On the other hand:

The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not natural. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid. Otherwise, one must choose a way to view each

in terms of a single group, and this choice can be arbitrary. In our example from topology, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point

to each point

in the same path-connected component.

As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector spaces with one endomorphism is nontrivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup H of a group

yields an action of

on the set of cosets of