Semigroupoids and Groupoids
This chapter amounts to a concise, self-contained introduction to the theory of semigroupoids and groupoids, from an algebraic and topologic point of view. In particular, a multitude of examples are presented and analyzed. On the algebraic side, an altern
- PDF / 545,049 Bytes
- 49 Pages / 439.36 x 666.15 pts Page_size
- 46 Downloads / 179 Views
Semigroupoids and Groupoids
This chapter is devoted to surveying the algebraic fundamentals of semigroupoids and groupoids and to reviewing the properties exhibited by these structures which play a crucial role in our subsequent work. While various results pertaining to the basic theory of semigroupoids and groupoids are scattered in the literature, we found it difficult to identify a few comprehensive, readily accessible accounts which cover all aspects dealt with here. That being said, two useful references we wish to single out are the monographs [86, 101]. These contain brief chapters on the basics of groupoid theory, though the overall focus is on the role played by groupoids in indexing representations of operator algebras. The interested reader is also referred to the expository paper [23], which, among other things, gives a flavor of the role played by groupoids in A. Grothendieck’s work in algebraic geometry, G.W. Mackey’s work in ergodic theory, A. Connes’ work in noncommutative geometry, and which contains a wealth of references to earlier articles. It is primarily for this reason that we decided to make the presentation of the material in this chapter as self-contained, free of excessive jargon, and accessible to the nonexpert, as realistically possible. In the process, we contribute to the existing theory by clarifying certain aspects and by proving several new results of independent interest.
2.1 Algebraic Considerations This section amounts to a concise, self-contained introduction to the algebraic theory of semigroupoids (Sect. 2.1.1) and groupoids (Sect. 2.1.2).
D. Mitrea et al., Groupoid Metrization Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8397-9 2, © Springer Science+Business Media New York 2013
11
12
2 Semigroupoids and Groupoids
2.1.1 Semigroupoids We start with the following definition. Definition 2.1. Given a nonempty set G, a partially defined binary operation on G is a function W G ! G, where G is a subset of G G called the domain of . If a; b 2 G, then call a b meaningfully defined if .a; b/ 2 G. Next, we recall the concept of semigroupoid. Definition 2.2. A semigroupoid is a nonempty set G equipped with a partially defined binary operation on G that is associative in the following precise sense. Let a; b; c 2 G. If a b and b c are meaningfully defined, then .a b/ c and a .b c/ are meaningfully defined and equal. Moreover, if either of these last two expressions is meaningfully defined, then so is the other and, again, they are equal. The reader is alerted to the fact that other terms are occasionally used in the literature in place of semigroupoid, most notably half-groupoid and incomplete groupoid. Given a semigroupoid .G; /, the binary operation W G ! G can be thought of as a partially defined multiplication. In this setting, it is customary to introduce ˚ G .2/ WD .a; b/ 2 G G W a b is meaningfully defined
(2.1)
and refer to the latter as the set of composable pairs of the semigroupoid G. Several relevant examples of
Data Loading...
