a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when...

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...

In mathematics, a Lie groupoid is a groupoid where the set Ob {\displaystyle \operatorname {Ob} } of objects and the set Mor {\displaystyle \operatorname...

mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category...

categories fibered in groupoids comes from groupoid objects internal to a category C {\displaystyle {\mathcal {C}}} . So given a groupoid object x ⇉ t s y {\displaystyle...

constant sheaf. The fundamental groupoid of the singleton space is the trivial groupoid (a groupoid with one object * and one morphism Hom(*, *) = {...

binary operation. The word groupoid is used by many universal algebraists, but workers in category theory and related areas object strongly to this usage...

of category the only difference between groupoid and group is that a groupoid may have more than one object but the group must have only one. Consider...

algebras can be seen as a generalization of group objects to monoidal categories. Groupoid object Awodey, Steve (2010), Category Theory, Oxford University...

theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism...

stack in groupoids or a (2,1)-sheaf if it is also fibered in groupoids, meaning that its fibers (the inverse images of objects of C) are groupoids. Some...

double groupoid generalises the notion of groupoid and of category to a higher dimension. A double groupoid D is a higher-dimensional groupoid involving...

category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the source and the target of...

case of monoid objects in the categories of small categories or of groupoids. Instead the notion of group object in the category of groupoids turns out to...

thought of as a "many-object generalisation" of a Lie algebra. Lie algebroids play a similar same role in the theory of Lie groupoids that Lie algebras play...

central groupoids are defined by an equational identity, they form a variety of algebras in which the free objects are called free central groupoids. Free...

\to \mathbf {C} } that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism...

charts and the gluing maps are isometries. Recall that a groupoid consists of a set of objects G 0 {\displaystyle G_{0}} , a set of arrows G 1 {\displaystyle...

unique element. A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds...

"The groupoid model refutes uniqueness of identity proofs", in which they showed that intensional type theory had a model in the category of groupoids. This...

Brown "Topology and Groupoids" pdf available Gives an account of some categorical methods in topology, use the fundamental groupoid on a set of base points...

object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects...

numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1,...

a notion of homotopy: it is a unit interval object in the category of groupoids. The category of groupoids admits all colimits, and in particular all pushouts...

generally, it is an exponential object in the category of G-sets. The notion of group action can be encoded by the action groupoid G′ = G ⋉ X associated to the...

{\displaystyle T^{*}M} is not always integrable to a Lie groupoid. A symplectic groupoid is a Lie groupoid G ⇉ M {\displaystyle {\mathcal {G}}\rightrightarrows...

group An isotropy group is the group of isomorphisms from any object to itself in a groupoid.[dubious – discuss] An isotropy representation is a representation...

In mathematics, the Mathieu groupoid M13 is a groupoid acting on 13 points such that the stabilizer of each point is the Mathieu group M12. It was introduced...

be composed unless they share an endpoint. Thus one has the fundamental groupoid instead of the fundamental group, and this construction is functorial....

1. A list object over an object A of C is: an object LA, a morphism oA : 1 → LA, and a morphism sA : A × LA → LA such that for any object B of C with...