Semigroup and Category-Theoretic Approaches to Partial Symmetry
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Sage 9.4 Reference Manual: Monoids Release 9.4
Sage 9.4 Reference Manual: Monoids Release 9.4 The Sage Development Team Aug 24, 2021 CONTENTS 1 Monoids 3 2 Free Monoids 5 3 Elements of Free Monoids 9 4 Free abelian monoids 11 5 Abelian Monoid Elements 15 6 Indexed Monoids 17 7 Free String Monoids 23 8 String Monoid Elements 29 9 Utility functions on strings 33 10 Hecke Monoids 35 11 Automatic Semigroups 37 12 Module of trace monoids (free partially commutative monoids). 47 13 Indices and Tables 55 Python Module Index 57 Index 59 i ii Sage 9.4 Reference Manual: Monoids, Release 9.4 Sage supports free monoids and free abelian monoids in any finite number of indeterminates, as well as free partially commutative monoids (trace monoids). CONTENTS 1 Sage 9.4 Reference Manual: Monoids, Release 9.4 2 CONTENTS CHAPTER ONE MONOIDS class sage.monoids.monoid.Monoid_class(names) Bases: sage.structure.parent.Parent EXAMPLES: sage: from sage.monoids.monoid import Monoid_class sage: Monoid_class(('a','b')) <sage.monoids.monoid.Monoid_class_with_category object at ...> gens() Returns the generators for self. EXAMPLES: sage: F.<a,b,c,d,e>= FreeMonoid(5) sage: F.gens() (a, b, c, d, e) monoid_generators() Returns the generators for self. EXAMPLES: sage: F.<a,b,c,d,e>= FreeMonoid(5) sage: F.monoid_generators() Family (a, b, c, d, e) sage.monoids.monoid.is_Monoid(x) Returns True if x is of type Monoid_class. EXAMPLES: sage: from sage.monoids.monoid import is_Monoid sage: is_Monoid(0) False sage: is_Monoid(ZZ) # The technical math meaning of monoid has ....: # no bearing whatsoever on the result: it's ....: # a typecheck which is not satisfied by ZZ ....: # since it does not inherit from Monoid_class. -
Nonablian Cocycles and Their Σ-Model Qfts
Nonablian cocycles and their σ-model QFTs December 31, 2008 Abstract Nonabelian cohomology can be regarded as a generalization of group cohomology to the case where both the group itself as well as the coefficient object are allowed to be generalized to 1-groupoids or even to general 1-categories. Cocycles in nonabelian cohomology in particular represent higher principal bundles (gerbes) { possibly equivariant, possibly with connection { as well as the corresponding associated higher vector bundles. We propose, expanding on considerations in [13, 34, 5], a systematic 1-functorial formalization of the σ-model quantum field theory associated with a given nonabelian cocycle regarded as the background field for a brane coupled to it. We define propagation in these σ-model QFTs and recover central aspects of groupoidification [1, 2] of linear algebra. In a series of examples we show how this formalization reproduces familiar structures in σ-models with finite target spaces such as Dijkgraaf-Witten theory and the Yetter model. The generalization to σ-models with smooth target spaces is developed elsewhere [24]. 1 Contents 1 Introduction 3 2 1-Categories and Homotopy Theory 4 2.1 Shapes for 1-cells . 5 2.2 !-Categories . 5 2.3 !-Groupoids . 7 2.4 Cosimplicial !-categories . 7 2.5 Monoidal biclosed structure on !Categories ............................ 7 2.6 Model structure on !Categories ................................... 9 3 Nonabelian cohomology and higher vector bundles 9 3.1 Principal !-bundles . 10 3.2 Associated !-bundles . 11 3.3 Sections and homotopies . 12 4 Quantization of !-bundles to σ-models 15 4.1 σ-Models . 16 4.2 Branes and bibranes . -