# variable groupoid

\xyoptioncurve

###### Definition 0.1.

A *variable groupoid* is defined as a family of groupoids
$\{{\U0001d5a6}_{\lambda}\}$ indexed by a parameter $\lambda \in T$ , with $T$ being either an index set^{} or a class (which may be a time parameter, for *time-dependent or dynamic groupoids ^{}*). If $\lambda $ belongs to a set $M$, then we may consider simply a projection

^{}$\U0001d5a6\times M\u27f6M$, which is an example of a trivial fibration

^{}. More generally, one can consider a

*fibration of groupoids*$\U0001d5a6\hookrightarrow Z\u27f6M$ (Higgins and Mackenzie, 1990) as defining a non-trivial

*variable groupoid*.

Remarks
An indexed family or class of topological groupoids^{} $[{\U0001d5a6}_{i}]$ with $i\in I$ in the category^{} Grpd of groupoids
with additional axioms, rules, or properties of the underlying topological groupoids,
that specify an indexed family of topological groupoid homomorphisms^{} for each variable groupoid
structure^{}.

Besides systems modelled in terms of a *fibration of groupoids*,
one may consider a *multiple groupoid* defined as a set of $N$
groupoid structures, any distinct pair of which satisfy an
interchange law which can be formulated as follows.
There exists a unique expression with the following content:

$$\left[\begin{array}{cc}\hfill x\hfill & \hfill y\hfill \\ \hfill z\hfill & \hfill w\hfill \end{array}\right]\mathit{\hspace{1em}}\text{objectmargin}=0pt\text{xy}(0,4)*+=\mathrm{"}a\mathrm{"},(0,-2)*+{\text{}}i=\mathrm{"}b\mathrm{"},(7,4)*+j=\mathrm{"}c\mathrm{"}\text{ar}\mathrm{@}->\mathrm{"}a\mathrm{"};\mathrm{"}b\mathrm{"}\text{ar}\mathrm{@}->\mathrm{"}a\mathrm{"};\mathrm{"}c\mathrm{"}\text{endxy},$$ | (0.1) |

where $i$ and $j$ must be distinct for this concept to be well defined. This uniqueness can also be represented by the equation

$$(x{\circ}_{j}y){\circ}_{i}(z{\circ}_{j}w)=(x{\circ}_{i}z){\circ}_{j}(y{\circ}_{i}w).$$ | (0.2) |

Remarks
This illustrates the principle that a 2-dimensional formula^{} may be
more comprehensible than a linear one.

Brown and Higgins, 1981a, showed that certain multiple groupoids
equipped with an extra structure called *connections* were
equivalent^{} to another structure called a *crossed complex*
which had already occurred in homotopy theory. such as
*double, or multiple* groupoids (Brown, 2004; 2005).
For example, the notion of an *atlas* of structures should,
in principle, apply to a lot of interesting, topological and/or
algebraic, structures: groupoids, multiple groupoids, Heyting
algebras, $n$-valued logic algebras^{} and ${C}^{*}$-convolution
-algebras^{}. Such examples occur frequently in *Higher Dimensional Algebra ^{}*
(HDA).

Title | variable groupoid |

Canonical name | VariableGroupoid |

Date of creation | 2013-03-22 18:15:45 |

Last modified on | 2013-03-22 18:15:45 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 17 |

Author | bci1 (20947) |

Entry type | Definition |

Classification | msc 55U05 |

Classification | msc 55U35 |

Classification | msc 55U40 |

Classification | msc 18G55 |

Classification | msc 18B40 |

Synonym | variable topology |

Related topic | VariableCategory |

Related topic | HigherDimensionalAlgebra |

Related topic | GroupoidCDynamicalSystem |

Related topic | HDA |

Related topic | VariableTopology |

Related topic | HigherDimensionalAlgebraHDA |

Related topic | Supercategories3 |

Related topic | 2Category2 |

Defines | family of groupoids |

Defines | GroupoidCDynamicalSystem |