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...

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

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

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...

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

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

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(*, *) = {...

X\times G\to X,} we get the groupoid G {\displaystyle {\mathcal {G}}} (= a category whose morphisms are all invertible) where objects are elements of X {\displaystyle...

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...

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...

groupoid. Then the inertia groupoid Λ U {\displaystyle \Lambda U} is a grouoiud (= a category whose morphisms are all invertible) where the objects are...

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...

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...

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,...

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...

invertible arrows, or morphisms. Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds (or...

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...

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...

(4) every groupoid object in X is effective. Mathematics portal Bousfield localization Homotopy hypothesis – Hypothesis that the ∞-groupoids are equivalent...

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

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...

Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-groupoids are spaces. One version of the hypothesis was claimed to be proved in...

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

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...

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...

Analogous to the fundamental groupoid it is possible to get rid of the choice of a base point and to define a monodromy groupoid. Here we consider (homotopy...

to a different generalization of Lie groups, namely Lie groupoids, which are groupoid objects in the category of smooth manifolds with a further requirement...

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...