The general equivalence of categories

1975 ◽  
pp. 57-91
Author(s):  
Michel Lazard
Keyword(s):  
Equivalence Of Categories

Crossed Semimodules and Schreier Internal Categories in the Category of Monoids

1998 ◽  
Vol 5 (6) ◽  
pp. 575-581 ◽  
Author(s):  
A. Patchkoria
Keyword(s):  
Internal Category ◽  
Equivalence Of Categories ◽  
Crossed Modules

Abstract We introduce the notion of a Schreier internal category in the category of monoids and prove that the category of Schreier internal categories in the category of monoids is equivalent to the category of crossed semimodules. This extends a well-known equivalence of categories between the category of internal categories in the category of groups and the category of crossed modules.

On a characterization of finitely presented functors

1989 ◽  
Vol 425 (1869) ◽  
pp. 329-339
Keyword(s):  
Representation Type ◽  
Equivalence Of Categories ◽  
Quotient Category ◽  
Auslander Algebra ◽  
Finitely Presented

The concept of finitely presented functor was introduced by Auslander. Proposition 3.1 of Auslander & Reiten provides a way of dealing with the category of finitely presented functors, that seems concrete and easy to use, at least in some examples. The study of this category, using this particular line of thought, is the main purpose of this work. In §1 I recall some basic definitions and give the required notation. In §2 I state the theorem of Auslander & Reiten referred to above and give a new proof of this result. The first part of this proof is an immediate consequence of the theory developed by Green. In §3 I state and prove an unpublished theorem by J. A. Green and I introduce a new category I such that the category of finitely presented functors. mmod A , is equivalent to a quotient category I / J , where J is an ideal of I . In §4 I give some examples of properties of mmod A , stated and proved in terms of the category I , by using the equivalence of categories referred to in §3. In §5 I consider the particular case where A = A q = k -alg < z : z q = 0>, apply the results of previous sections to study mmod A q and make conclusions about the representation type of the Auslander algebra of A q .

scholarly journals Univalent categories and the Rezk completion

2015 ◽  
Vol 25 (5) ◽  
pp. 1010-1039 ◽  
Author(s):  
BENEDIKT AHRENS ◽  
KRZYSZTOF KAPULKIN ◽  
MICHAEL SHULMAN
Keyword(s):  
Category Theory ◽  
Type Theory ◽  
Dependent Type Theory ◽  
Equivalence Of Categories ◽  
Segal Spaces ◽  
Definition Of ◽  
Categorical Semantics ◽  
Univalent Foundations ◽  
Dependent Type

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of ‘category’ for which equality and equivalence of categories agree. Such categories satisfy a version of the univalence axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them ‘saturated’ or ‘univalent’ categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.

scholarly journals Quantum integrals and the affineness criterion for quantum Yetter-Drinfeld π-modules

Filomat ◽  
2012 ◽  
Vol 26 (1) ◽  
pp. 101-118
Author(s):  
Chen Quan-Guo ◽  
Wang Shuan-Hong
Keyword(s):  
Drinfeld Modules ◽  
Canonical Map ◽  
Equivalence Of Categories ◽  
Quantum Integral ◽  
Special Cases

In the paper, the quantum integrals associated to quantum Yetter-Drinfeld ?-modules are defined. We shall prove the following affineness criterion: if there exists ? = {?? : H? ? Hom(H?-1, A)} ? ? ? a total quantum integral and the canonical map ? : A?B A ? ???? H? ? A, ?(a?B b)= ???? S?-1 ??(b[1,?-1?-1?])b[0,0]<-1,?> ? ab[0,0]<0,0> is subjective. Then the induction functor -?B A : UB ? H YD?A is an equivalence of categories. The affineness criterion proven by Menini and Militaru is recovered as special cases.

scholarly journals CIRCLE ACTIONS, CENTRAL EXTENSIONS AND STRING STRUCTURES

2010 ◽  
Vol 07 (06) ◽  
pp. 1065-1092 ◽  
Author(s):  
MICHAEL K. MURRAY ◽  
RAYMOND F. VOZZO
Keyword(s):  
Differential Form ◽  
Central Extensions ◽  
Circle Actions ◽  
The Real ◽  
Circle Bundles ◽  
String Class ◽  
Equivalence Of Categories ◽  
Bundle Gerbes

The caloron correspondence can be understood as an equivalence of categories between G-bundles over circle bundles and LG ⋊ρ S1-bundles where LG is the group of smooth loops in G. We use it, and lifting bundle gerbes, to derive an explicit differential form based formula for the (real) string class of an LG ⋊ρ S1-bundle.

scholarly journals Complex manifolds with maximal torus actions

2019 ◽  
Vol 2019 (751) ◽  
pp. 121-184 ◽  
Author(s):  
Hiroaki Ishida
Keyword(s):  
Complete Classification ◽  
Vector Subspace ◽  
Toric Geometry ◽  
Complex Manifolds ◽  
Combinatorial Objects ◽  
Complex Dimension ◽  
Compact Torus ◽  
Equivalence Of Categories ◽  
Compact Tori

AbstractIn this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples {(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in {\mathfrak{g}}, complex vector subspace {\mathfrak{h}} of {\mathfrak{g}^{\mathbb{C}}} and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate {\mathbb{C}^{n}}-actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.

scholarly journals A categorical invariant of flow equivalence of shifts

2014 ◽  
Vol 36 (2) ◽  
pp. 470-513 ◽  
Author(s):  
ALFREDO COSTA ◽  
BENJAMIN STEINBERG
Keyword(s):  
Analogous Result ◽  
Mild Conditions ◽  
Equivalence Of Categories ◽  
Natural Equivalence ◽  
Sofic Shifts ◽  
Flow Equivalence ◽  
Flow Invariance ◽  
Natural Sense

We prove that the Karoubi envelope of a shift—defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift—is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants—some new and some old—are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov–Dyck and Markov–Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property ($\mathscr{A}$) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property ($\mathscr{A}$) is decidable for sofic shifts.

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