Physics Library
 An open source physics library
Encyclopedia | Forums | Docs | Random | Template Test |  
Login
create new user
Username:
Password:
forget your password?
Main Menu
Sections

Talkback

Downloads

Information
category of Borel spaces (Definition)
Definition 0.1   A category of Borel spaces $\mathbb{B}$ has, as its objects, all Borel spaces $(X_b;\mathcal{B}(X_b))$, and as its morphisms the Borel morphisms $f_b$ between Borel spaces; the Borel morphism composition is defined so that it preserves the Borel structure determined by the $\sigma$-algebra of Borel sets.
Remark 0.1   The category of standard Borel G-spaces $\mathbb{B}_G$ is defined in a similar manner to $\mathbb{B}$, with the additional condition that Borel G-space morphisms commute with the Borel actions $a: G \times X \to X$ defined as Borel functions (or Borel-measurable maps). Thus, $\mathbb{B}_G$ is a subcategory of $\mathbb{B}$; in its turn, $\mathbb{B}$ is a subcategory of $\mathbb{T}op$–the category of topological spaces and continuous functions.

The category of rigid Borel spaces can be defined as above with the additional condition that the only automorphism $f: X_b \to X_b$ (bijection) is the identity $1_{(X_b; \mathcal{B}(X_b))}$.



"category of Borel spaces" is owned by bci1.

View style:

Keywords:  category of Borel spaces

Cross-references: identity, functions, topological, Borel-measurable maps, Borel actions, commute, Borel G-space, G-spaces, category, Borel sets, composition, Borel morphisms, morphisms, Borel spaces, objects
There are 3 references to this object.

This is version 2 of category of Borel spaces, born on 2009-02-04, modified 2009-03-03.
Object id is 495, canonical name is CategoryOfBorelSpaces.
Accessed 371 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:

No messages.

Testing some escape charachters for html category with a generator has an injective cogenerator" now escape ” with "