Spectral Algebra Models of Unstable V N-Periodic Homotopy Theory
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HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition. -
Arxiv:1906.03655V2 [Math.AT] 1 Jul 2020
RATIONAL HOMOTOPY EQUIVALENCES AND SINGULAR CHAINS MANUEL RIVERA, FELIX WIERSTRA, MAHMOUD ZEINALIAN Abstract. Bousfield and Kan’s Q-completion and fiberwise Q-completion of spaces lead to two different approaches to the rational homotopy theory of non-simply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of path-connected pointed spaces is a weak equivalence if it induces an isomorphism between fun- damental groups and higher rationalized homotopy groups; we call these maps π1-rational homotopy equivalences. In this paper, we compare these two notions and show that π1-rational homotopy equivalences correspond to maps that induce Ω-quasi-isomorphisms on the rational singular chains, i.e. maps that induce a quasi-isomorphism after applying the cobar functor to the dg coassociative coalge- bra of rational singular chains. This implies that both notions of rational homotopy equivalence can be deduced from the rational singular chains by using different alge- braic notions of weak equivalences: quasi-isomorphism and Ω-quasi-isomorphisms. We further show that, in the second approach, there are no dg coalgebra models of the chains that are both strictly cocommutative and coassociative. 1. Introduction One of the questions that gave birth to rational homotopy theory is the commuta- tive cochains problem which, given a commutative ring k, asks whether there exists a commutative differential graded (dg) associative k-algebra functorially associated to any topological space that is weakly equivalent to the dg associative algebra of singu- lar k-cochains on the space with the cup product [S77], [Q69]. -
Arxiv:0806.3256V2 [Math.RT] 12 Dec 2008 Vector a Sdsrbn Nafiespace Affine an Describing As Ment 1.1
GALE DUALITY AND KOSZUL DUALITY TOM BRADEN, ANTHONY LICATA, NICHOLAS PROUDFOOT, AND BEN WEBSTER ABSTRACT. Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O. 1. INTRODUCTION In this paper we define and study a class of finite-dimensional graded algebras which are related to the combinatorics of hyperplane arrangements and to the ge- ometry of hypertoric varieties. The categories of representations of these algebras are similar in structure to the integral blocks of category O, originally introduced by Bernstein-Gelfand-Gelfand [BGG76]. Our categories share many important proper- ties with such blocks, including a highest weight structure, a Koszul grading, and a relationship with the geometry of a symplectic variety. As with category O, there is a nice interpretation of Koszul duality in our setting, and there are interesting families of functors between our categories. In this paper we take a combinatorial approach, analogous to that taken by Stroppel in her study of category O [Str03]. In a subsequent paper [BLPWa] we will take an approach to these categories more analogous to the classical perspective on category O; we will realize them as cate- arXiv:0806.3256v2 [math.RT] 12 Dec 2008 gories of modules over an infinite dimensional algebra, and as a certain category of sheaves on a hypertoric variety, related by a localization theorem extending that of Beilinson-Bernstein [BB81]. -
Langlands' Philosophy and Koszul Duality
LANGLANDS PHILOSOPHY AND KOSZUL DUALITY WOLFGANG SOERGEL Contents Introduction The basic conjecture The easiest example The geometry of AdamsBarbaschVogan Thanks Notations and conventions Groups and varieties Grothendieck groups Equivariant derived categories Geometric extension algebras Examples Tori Generic central character The case G SL Complex groups Compact groups Tempered representations The center and geometry Further conjectures Generalities on Koszul duality Application to our situation Motivation References Introduction In this article I will formulate conjectures relating representation theory to geometry and prove them in some cases I am trying to extend work of AdamsBarbaschVogan ABV and some joint work of myself with Beilinson and Ginzburg BGS All this should b e regarded as a contribution to the lo cal case of Langlands philosophy SOERGEL The lo calization of BeilinsonBernstein also establishes a relation b etween representation theory and geometry This is however very dierent from the relation to b e investigated in this article Whereas lo calization leads to geometry on the group itself or its ag manifold the results in ABV to b e extended in this article lead to geometry on the Langlands dual group The basic conjecture Let G b e only for a short moment a real reductive Zariskiconnected algebraic group We want to study admissible representations of the asso ciated Lie group GR of real p oints of G More precisely we want to study the category MGR of smo oth admissible -
Rational Homotopy Theory: a Brief Introduction
Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article. -
The Continuum As a Final Coalgebra
Theoretical Computer Science 280 (2002) 105–122 www.elsevier.com/locate/tcs The continuum as a ÿnal coalgebra D. PavloviÃca, V. Prattb; ∗ aKestrel Institute, Palo Alto, CA, USA bDepartment of Computer Science, Stanford University, Stanford, CA 94305-9045, USA Abstract We deÿne the continuum up to order isomorphism, and hence up to homeomorphism via the order topology, in terms of the ÿnal coalgebra of either the functor N ×X , product with the set of natural numbers, or the functor 1+N ×X . This makes an attractive analogy with the deÿnition of N itself as the initial algebra of the functor 1 + X , disjoint union with a singleton. We similarly specify Baire space and Cantor space in terms of these ÿnal coalgebras. We identify two variants of this approach, a coinductive deÿnition based on ÿnal coalgebraic structure in the category of sets, and a direct deÿnition as a ÿnal coalgebra in the category of posets. We conclude with some paradoxical discrepancies between continuity and constructiveness in this setting. c 2002 Elsevier Science B.V. All rights reserved. 1. Introduction The algebra (N; 0;s) of non-negative integers under successor s with constant 0, the basic number system of mathematics, occupies a distinguished position in the category of algebras having a unary operation and a constant, namely as its initial object. In this paper we show that the continuum, the basic environment of physics, occu- pies an equally distinguished position in the category of coalgebras having one “dual operation” of arity N, namely as its ÿnal object. Here a dual operation is understood as a function X → X + X + ··· dual to an operation as a function X × X ×···X , with arity being the number of copies of X in each case. -
Arxiv:1907.08255V2 [Math.RA] 26 Aug 2020 Olers Eso Htte R Pitn of Splitting Are They That Show We Coalgebras
COHOMOLOGY AND DEFORMATIONS OF DENDRIFORM COALGEBRAS APURBA DAS Abstract. Dendriform coalgebras are the dual notion of dendriform algebras and are splitting of as- sociative coalgebras. In this paper, we define a cohomology theory for dendriform coalgebras based on some combinatorial maps. We show that the cohomology with self coefficients governs the formal defor- mation of the structure. We also relate this cohomology with the cohomology of dendriform algebras, coHochschild (Cartier) cohomology of associative coalgebras and cohomology of Rota-Baxter coalgebras which we introduce in this paper. Finally, using those combinatorial maps, we introduce homotopy analogue of dendriform coalgebras and study some of their properties. 1. Introduction Dendriform algebras were first introduced by Jean-Louis Loday in his study on periodicity phe- nomenons in algebraic K-theory [13]. These algebras are Koszul dual to diassociative algebras (also called associative dialgebras). More precisely, a dendriform algebra is a vector space equipped with two binary operations satisfying three new identities. The sum of the two operations turns out to be associa- tive. Thus, a dendriform algebra can be thought of like a splitting of associative algebras. Dendriform algebras are closely related to Rota-Baxter algebras [1,8]. Recently, the present author defines a coho- mology and deformation theory for dendriform algebras [3]. See [6–8,17] and references therein for more literature about dendriform algebras. The dual picture of dendriform algebras is given by dendriform coalgebras. It is given by two coproducts on a vector space satisfying three identities (dual to dendriform algebra identities), and the sum of the two coproducts makes the underlying vector space into an associative coalgebra. -
The Method of Coalgebra: Exercises in Coinduction
The Method of Coalgebra: exercises in coinduction Jan Rutten CWI & RU [email protected] Draft d.d. 8 July 2018 (comments are welcome) 2 Draft d.d. 8 July 2018 Contents 1 Introduction 7 1.1 The method of coalgebra . .7 1.2 History, roughly and briefly . .7 1.3 Exercises in coinduction . .8 1.4 Enhanced coinduction: algebra and coalgebra combined . .8 1.5 Universal coalgebra . .8 1.6 How to read this book . .8 1.7 Acknowledgements . .9 2 Categories { where coalgebra comes from 11 2.1 The basic definitions . 11 2.2 Category theory in slogans . 12 2.3 Discussion . 16 3 Algebras and coalgebras 19 3.1 Algebras . 19 3.2 Coalgebras . 21 3.3 Discussion . 23 4 Induction and coinduction 25 4.1 Inductive and coinductive definitions . 25 4.2 Proofs by induction and coinduction . 28 4.3 Discussion . 32 5 The method of coalgebra 33 5.1 Basic types of coalgebras . 34 5.2 Coalgebras, systems, automata ::: ....................... 34 6 Dynamical systems 37 6.1 Homomorphisms of dynamical systems . 38 6.2 On the behaviour of dynamical systems . 41 6.3 Discussion . 44 3 4 Draft d.d. 8 July 2018 7 Stream systems 45 7.1 Homomorphisms and bisimulations of stream systems . 46 7.2 The final system of streams . 52 7.3 Defining streams by coinduction . 54 7.4 Coinduction: the bisimulation proof method . 59 7.5 Moessner's Theorem . 66 7.6 The heart of the matter: circularity . 72 7.7 Discussion . 76 8 Deterministic automata 77 8.1 Basic definitions . 78 8.2 Homomorphisms and bisimulations of automata . -
Curved Koszul Duality for Algebras Over Unital Operads Najib Idrissi
Curved Koszul Duality for Algebras over Unital Operads Najib Idrissi To cite this version: Najib Idrissi. Curved Koszul Duality for Algebras over Unital Operads. 2018. hal-01786218 HAL Id: hal-01786218 https://hal.archives-ouvertes.fr/hal-01786218 Preprint submitted on 5 May 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Curved Koszul Duality for Algebras over Unital Operads Najib Idrissi∗ May 4, 2018 We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over binary unital operads. As an ap- plication, we study Poisson n-algebras given by polynomial functions on a standard shifted symplectic space. We compute explicit resolutions of these algebras using curved Koszul duality. We use these resolutions to compute derived enveloping algebras and factorization homology on parallelized simply connected closed manifolds of these Poisson n-algebras. Contents Introduction1 1 Conventions, background, and recollections4 2 Curved coalgebras, semi-augmented algebras, bar-cobar adjunction9 3 Koszul duality of unitary algebras 14 4 Application: symplectic Poisson n-algebras 18 Introduction Koszul duality was initially developed by Priddy [Pri70] for associative algebras. Given an augmented associative algebra A, there is a “Koszul dual” algebra A!, and there is an equivalence (subject to some conditions) between parts of the derived categories of A and A!. -
Coalgebras from Formulas
Coalgebras from Formulas Serban Raianu California State University Dominguez Hills Department of Mathematics 1000 E Victoria St Carson, CA 90747 e-mail:[email protected] Abstract Nichols and Sweedler showed in [5] that generic formulas for sums may be used for producing examples of coalgebras. We adopt a slightly different point of view, and show that the reason why all these constructions work is the presence of certain representative functions on some (semi)group. In particular, the indeterminate in a polynomial ring is a primitive element because the identity function is representative. Introduction The title of this note is borrowed from the title of the second section of [5]. There it is explained how each generic addition formula naturally gives a formula for the action of the comultiplication in a coalgebra. Among the examples chosen in [5], this situation is probably best illus- trated by the following two: Let C be a k-space with basis {s, c}. We define ∆ : C −→ C ⊗ C and ε : C −→ k by ∆(s) = s ⊗ c + c ⊗ s ∆(c) = c ⊗ c − s ⊗ s ε(s) = 0 ε(c) = 1. 1 Then (C, ∆, ε) is a coalgebra called the trigonometric coalgebra. Now let H be a k-vector space with basis {cm | m ∈ N}. Then H is a coalgebra with comultiplication ∆ and counit ε defined by X ∆(cm) = ci ⊗ cm−i, ε(cm) = δ0,m. i=0,m This coalgebra is called the divided power coalgebra. Identifying the “formulas” in the above examples is not hard: the for- mulas for sin and cos applied to a sum in the first example, and the binomial formula in the second one. -
Rational Homotopy Theory of Mapping Spaces Via Lie Theory for L-Infinity
RATIONAL HOMOTOPY THEORY OF MAPPING SPACES VIA LIE THEORY FOR L∞-ALGEBRAS ALEXANDER BERGLUND Abstract. We calculate the higher homotopy groups of the Deligne-Getzler ∞-groupoid associated to a nilpotent L∞-algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces. 1. Introduction In [17] Getzler associates an ∞-groupoid γ•(g) to a nilpotent L∞-algebra g, which generalizes the Deligne groupoid of a nilpotent differential graded Lie algebra [16, 17, 19, 24]. It is well known that the set of path components π0γ•(g) may be identified with the ‘moduli space’ M C (g) of equivalence classes of Maurer-Cartan elements, which plays an important role in deformation theory. Our main result is the calculation of the higher homotopy groups. Theorem 1.1. For a nilpotent L∞-algebra g there is an explicitly defined natural group isomorphism τ B : Hn(g ) → πn+1(γ•(g), τ), n ≥ 0, τ where g denotes the L∞-algebra g twisted by the Maurer-Cartan element τ. The τ group structure on H0(g ) is given by the Campbell-Hausdorff formula. As a corollary, we obtain a characterization of when an L∞-morphism induces an equivalence between the associated ∞-groupoids. Corollary 1.2. An L∞-morphism between nilpotent L∞-algebras f : g → h in- duces an equivalence of ∞-groupoids γ•(g) → γ•(h) if and only if the following two conditions are satisfied: (1) The map on Maurer-Cartan moduli M C (g) → M C (h) is a bijection. arXiv:1110.6145v2 [math.AT] 1 Aug 2015 τ τ f∗(τ) (2) The L∞-morphism f : g → h induces an isomorphism in homology in non-negative degrees, for every Maurer-Cartan element τ in g. -
GALE DUALITY and KOSZUL DUALITY in This Paper We Define
GALE DUALITY AND KOSZUL DUALITY TOM BRADEN, ANTHONY LICATA, NICHOLAS PROUDFOOT, AND BEN WEBSTER ABSTRACT. Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O. 1. INTRODUCTION In this paper we define and study a class of finite-dimensional graded algebras which are related to the combinatorics of hyperplane arrangements and to the ge- ometry of hypertoric varieties. The categories of representations of these algebras are similar in structure to the integral blocks of category O, originally introduced by Bernstein-Gelfand-Gelfand [BGG76]. Our categories share many important proper- ties with such blocks, including a highest weight structure, a Koszul grading, and a relationship with the geometry of a symplectic variety. As with category O, there is a nice interpretation of Koszul duality in our setting, and there are interesting families of functors between our categories. In this paper we take a combinatorial approach, analogous to that taken by Stroppel in her study of category O [Str03]. In a subsequent paper [BLPWa] we will take an approach to these categories more analogous to the classical perspective on category O; we will realize them as cate- gories of modules over an infinite dimensional algebra, and as a certain category of sheaves on a hypertoric variety, related by a localization theorem extending that of Beilinson-Bernstein [BB81].