Liftable Derivations for Generically Separably Algebraic Morphisms of Schemes
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On String Topology Operations and Algebraic Structures on Hochschild Complexes
City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 9-2015 On String Topology Operations and Algebraic Structures on Hochschild Complexes Manuel Rivera Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/1107 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] On String Topology Operations and Algebraic Structures on Hochschild Complexes by Manuel Rivera A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York 2015 c 2015 Manuel Rivera All Rights Reserved ii This manuscript has been read and accepted for the Graduate Faculty in Mathematics in sat- isfaction of the dissertation requirements for the degree of Doctor of Philosophy. Dennis Sullivan, Chair of Examining Committee Date Linda Keen, Executive Officer Date Martin Bendersky Thomas Tradler John Terilla Scott Wilson Supervisory Committee THE CITY UNIVERSITY OF NEW YORK iii Abstract On string topology operations and algebraic structures on Hochschild complexes by Manuel Rivera Adviser: Professor Dennis Sullivan The field of string topology is concerned with the algebraic structure of spaces of paths and loops on a manifold. It was born with Chas and Sullivan’s observation of the fact that the in- tersection product on the homology of a smooth manifold M can be combined with the con- catenation product on the homology of the based loop space on M to obtain a new product on the homology of LM , the space of free loops on M . -
On Projective Lift and Orbit Spaces T.K
BULL. AUSTRAL. MATH. SOC. 54GO5, 54H15 VOL. 50 (1994) [445-449] ON PROJECTIVE LIFT AND ORBIT SPACES T.K. DAS By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category d of regular Hausdorff spaces and continuous dp- epimorphisms to its coreflective subcategory £d consisting of projective objects of Cd • We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd • 1. INTRODUCTION Throughout the paper all spaces are regular HausdorfF and maps are continuous epimorphisms. By X, Y we denote spaces and for A C X, Cl A and Int A mean the closure of A and the interior of A in X respectively. The complete Boolean algebra of regular closed sets of a space X is denoted by R(X) and the Stone space of R(X) by S(R(X)). The pair (EX, hx) is the projective cover of X, where EX is the subspace of S(R(X)) having convergent ultrafilters as its members and hx is the natural map from EX to X, sending T to its point of convergence ^\T (a singleton is identified with its member). We recall here that {fl(F) \ F £ R(X)} is an open base for the topology on EX, where d(F) = {T £ EX \ F G T} [8]. The projective lift of a map /: X —> Y (if it exists) is the unique map Ef: EX —> EY satisfying hy o Ef = f o hx • In [4], Henriksen and Jerison showed that when X , Y are compact spaces, then the projective lift Ef of / exists if and only if / satisfies the following condition, hereafter called the H J - condition, holds: Clint/-1(F) = Cl f-^IntF), for each F in R(Y). -
ADJOINT DIVISORS and FREE DIVISORS Contents 1. Introduction
ADJOINT DIVISORS AND FREE DIVISORS DAVID MOND AND MATHIAS SCHULZE Abstract. We describe two situations where adding the adjoint divisor to a divisor D with smooth normalization yields a free divisor. Both also involve stability or versality. In the first, D is the image of a corank 1 stable map-germ (Cn, 0) → (Cn+1, 0), and is not free. In the second, D is the discriminant of a versal deformation of a weighted homogeneous function with isolated critical point (subject to certain numerical conditions on the weights). Here D itself is already free. We also prove an elementary result, inspired by these first two, from which we obtain a plethora of new examples of free divisors. The presented results seem to scratch the surface of a more general phenomenon that is still to be revealed. Contents 1. Introduction 1 Notation 4 Acknowledgments 4 2. Images of stable maps 4 3. Discriminants of hypersurface singularities 12 4. Pull-back of free divisors 20 References 22 1. Introduction Let M be an n-dimensional complex analytic manifold and D a hypersurface in M. The OM -module Der( log D) of logarithmic vector fields along D consists of all vector fields on M tangent− to D at all smooth points of D. If this module is locally free, D is called a free divisor. This terminology was introduced by Kyoji Saito in [Sai80b]. As freeness is evidently a local condition, so we may pass to n germs of analytic spaces D X := (C , 0), pick coordinates x1,...,xn on X and a defining equation h O ⊂= C x ,...,x for D. -
Parametrized Higher Category Theory
Parametrized higher category theory Jay Shah MIT May 1, 2017 Jay Shah (MIT) Parametrized higher category theory May 1, 2017 1 / 32 Answer: depends on the class of weak equivalences one inverts in the larger category of G-spaces. Inverting the class of maps that induce a weak equivalence of underlying spaces, X ; the homotopy type of the underlying space X , together with the homotopy coherent G-action. Can extract homotopy fixed points and hG orbits X , XhG from this. Equivariant homotopy theory Let G be a finite group and let X be a topological space with G-action (e.g. G = C2 and X = U(n) with the complex conjugation action). What is the \homotopy type" of X ? Jay Shah (MIT) Parametrized higher category theory May 1, 2017 2 / 32 Inverting the class of maps that induce a weak equivalence of underlying spaces, X ; the homotopy type of the underlying space X , together with the homotopy coherent G-action. Can extract homotopy fixed points and hG orbits X , XhG from this. Equivariant homotopy theory Let G be a finite group and let X be a topological space with G-action (e.g. G = C2 and X = U(n) with the complex conjugation action). What is the \homotopy type" of X ? Answer: depends on the class of weak equivalences one inverts in the larger category of G-spaces. Jay Shah (MIT) Parametrized higher category theory May 1, 2017 2 / 32 Equivariant homotopy theory Let G be a finite group and let X be a topological space with G-action (e.g. -
Lifting Grothendieck Universes
Lifting Grothendieck Universes Martin HOFMANN, Thomas STREICHER Fachbereich 4 Mathematik, TU Darmstadt Schlossgartenstr. 7, D-64289 Darmstadt mh|[email protected] Spring 1997 Both in set theory and constructive type theory universes are a useful and necessary tool for formulating abstract mathematics, e.g. when one wants to quantify over all structures of a certain kind. Structures of a certain kind living in universe U are usually referred to as \small structures" (of that kind). Prominent examples are \small monoids", \small groups" . and last, but not least \small sets". For (classical) set theory an appropriate notion of universe was introduced by A. Grothendieck for the purposes of his development of Grothendieck toposes (of sheaves over a (small) site). Most concisely, a Grothendieck universe can be defined as a transitive set U such that (U; 2 U×U ) itself constitutes a model of set theory.1 In (constructive) type theory a universe (in the sense of Martin–L¨of) is a type U of types that is closed under the usual type forming operations as e.g. Π, Σ and Id. More precisely, it is a type U together with a family of types ( El(A) j A 2 U ) assigning its type El(A) to every A 2 U. Of course, El(A) = El(B) iff A = B 2 U. For the purposes of Synthetic Domain Theory (see [6, 3]) or Semantic Nor- malisations Proofs (see [1]) it turns out to be necessary to organise (pre)sheaf toposes into models of type theory admitting a universe. In this note we show how a Grothendieck universe U gives rise to a type-theoretic universe in the presheaf topos Cb where C is a small category (i.e. -
Arxiv:1709.06839V3 [Math.AT] 8 Jun 2021
PRODUCT AND COPRODUCT IN STRING TOPOLOGY NANCY HINGSTON AND NATHALIE WAHL Abstract. Let M be a closed Riemannian manifold. We extend the product of Goresky- Hingston, on the cohomology of the free loop space of M relative to the constant loops, to a nonrelative product. It is graded associative and commutative, and compatible with the length filtration on the loop space, like the original product. We prove the following new geometric property of the dual homology coproduct: the nonvanishing of the k{th iterate of the coproduct on a homology class ensures the existence of a loop with a (k + 1){fold self-intersection in every representative of the class. For spheres and projective spaces, we show that this is sharp, in the sense that the k{th iterated coproduct vanishes precisely on those classes that have support in the loops with at most k{fold self-intersections. We study the interactions between this cohomology product and the more well-known Chas-Sullivan product. We give explicit integral chain level constructions of these loop products and coproduct, including a new construction of the Chas- Sullivan product, which avoid the technicalities of infinite dimensional tubular neighborhoods and delicate intersections of chains in loop spaces. Let M be a closed oriented manifold of dimension n, for which we pick a Riemannian metric, and let ΛM = Maps(S1;M) denote its free loop space. Goresky and the first author defined in [15] a ∗ product ~ on the cohomology H (ΛM; M), relative to the constant loops M ⊂ ΛM. They showed that this cohomology product behaves as a form of \dual" of the Chas-Sullivan product [8] on the homology H∗(ΛM), at least through the eyes of the energy functional. -
Arxiv:2007.12573V3 [Math.AC] 28 Aug 2020 1,Tm .] Egtamr Rcs Eso Fteegnau Theore Eigenvalue the of Version Precise 1.2 More a Theorem Get We 3.3], Thm
STICKELBERGER AND THE EIGENVALUE THEOREM DAVID A. COX To David Eisenbud on the occasion of his 75th birthday. Abstract. This paper explores the relation between the Eigenvalue Theorem and the work of Ludwig Stickelberger (1850-1936). 1. Introduction The Eigenvalue Theorem is a standard result in computational algebraic geom- etry. Given a field F and polynomials f1,...,fs F [x1,...,xn], it is well known that the system ∈ (1.1) f = = fs =0 1 ··· has finitely many solutions over the algebraic closure F of F if and only if A = F [x ,...,xn]/ f ,...,fs 1 h 1 i has finite dimension over F (see, for example, Theorem 6 of [7, Ch. 5, 3]). § A polynomial f F [x ,...,xn] gives a multiplication map ∈ 1 mf : A A. −→ A basic version of the Eigenvalue Theorem goes as follows: Theorem 1.1 (Eigenvalue Theorem). When dimF A< , the eigenvalues of mf ∞ are the values of f at the finitely many solutions of (1.1) over F . For A = A F F , we have a canonical isomorphism of F -algebras ⊗ A = Aa, ∈V a F Y(f1,...,fs) where Aa is the localization of A at the maximal ideal corresponding to a. Following [12, Thm. 3.3], we get a more precise version of the Eigenvalue Theorem: arXiv:2007.12573v3 [math.AC] 28 Aug 2020 Theorem 1.2 (Stickelberger’s Theorem). For every a V (f ,...,fs), we have ∈ F 1 mf (Aa) Aa, and the restriction of mf to Aa has only one eigenvalue f(a). ⊆ This result easily implies Theorem 1.1 and enables us to compute the character- istic polynomial of mf . -
On Closed Categories of Functors
ON CLOSED CATEGORIES OF FUNCTORS Brian Day Received November 7, 19~9 The purpose of the present paper is to develop in further detail the remarks, concerning the relationship of Kan functor extensions to closed structures on functor categories, made in "Enriched functor categories" | 1] §9. It is assumed that the reader is familiar with the basic results of closed category theory, including the representation theorem. Apart from some minor changes mentioned below, the terminology and notation employed are those of |i], |3], and |5]. Terminology A closed category V in the sense of Eilenberg and Kelly |B| will be called a normalised closed category, V: V o ÷ En6 being the normalisation. Throughout this paper V is taken to be a fixed normalised symmetric monoidal closed category (Vo, @, I, r, £, a, c, V, |-,-|, p) with V ° admitting all small limits (inverse limits) and colimits (direct limits). It is further supposed that if the limit or colimit in ~o of a functor (with possibly large domain) exists then a definite choice has been made of it. In short, we place on V those hypotheses which both allow it to replace the cartesian closed category of (small) sets Ens as a ground category and are satisfied by most "natural" closed categories. As in [i], an end in B of a V-functor T: A°P@A ÷ B is a Y-natural family mA: K ÷ T(AA) of morphisms in B o with the property that the family B(1,mA): B(BK) ÷ B(B,T(AA)) in V o is -2- universally V-natural in A for each B 6 B; then an end in V turns out to be simply a family sA: K ~ T(AA) of morphisms in V o which is universally V-natural in A. -
Mor04, Mor12], Provides a Foundational Tool for 1 Solving Problems in A1-Enumerative Geometry
THE TRACE OF THE LOCAL A1-DEGREE THOMAS BRAZELTON, ROBERT BURKLUND, STEPHEN MCKEAN, MICHAEL MONTORO, AND MORGAN OPIE Abstract. We prove that the local A1-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local A1-degree over the residue field. This fact was originally suggested by Morel's work on motivic transfers, and by Kass and Wickelgren's work on the Scheja{Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren relating the Scheja{Storch form and the local A1-degree. 1. Introduction The A1-degree, first defined by Morel [Mor04, Mor12], provides a foundational tool for 1 solving problems in A1-enumerative geometry. In contrast to classical notions of degree, 1 n n the local A -degree is not integer valued: given a polynomial function f : Ak ! Ak with 1 1 A isolated zero p, the local A -degree of f at p, denoted by degp (f), is defined to be an element of the Grothendieck{Witt group of the ground field. Definition 1.1. Let k be a field. The Grothendieck{Witt group GW(k) is defined to be the group completion of the monoid of isomorphism classes of symmetric non-degenerate bilinear forms over k. The group operation is the direct sum of bilinear forms. We may also give GW(k) a ring structure by taking tensor products of bilinear forms for our multiplication. The local A1-degree, which will be defined in Definition 2.9, can be related to other important invariants at rational points. -
Adjunctions of Higher Categories
Adjunctions of Higher Categories Joseph Rennie 3/1/2017 Adjunctions arise very naturally in many diverse situations. Moreover, they are a primary way of studying equivalences of categories. The goal of todays talk is to discuss adjunctions in the context of higher category theories. We will give two definitions of an adjunctions and indicate how they have the same data. From there we will move on to discuss standard facts that hold in adjunctions of higher categories. Most of what we discuss can be found in section 5.2 of [Lu09], particularly subsection 5.2.2. Adjunctions of Categories Recall that for ordinary categories C and D we defined adjunctions as follows Definition 1.1. (Definition Section IV.1 Page 78 [Ml98]) Let F : C ! D and G : D ! C be functors. We say the pair (F; G) is an adjunction and denote it as F C D G if for each object C 2 C and D 2 D there exists an equivalence ∼ HomD(F C; D) = HomC(C; GD) This definition is not quite satisfying though as the determination of the iso- morphisms is not functorial here. So, let us give following equivalent definitions that are more satisfying: Theorem 1.2. (Parts of Theorem 2 Section IV.1 Page 81 [Ml98]) A pair (F; G) is an adjunction if and only if one of the following equivalent conditions hold: 1. There exists a natural transformation c : FG ! idD (the counit map) such that the natural map F (cC )∗ HomC(C; GD) −−! HomD(F C; F GD) −−−−−! HomD(F C; D) is an isomorphism 1 2. -
All (,1)-Toposes Have Strict Univalent Universes
All (1; 1)-toposes have strict univalent universes Mike Shulman University of San Diego HoTT 2019 Carnegie Mellon University August 13, 2019 One model is not enough A (Grothendieck{Rezk{Lurie)( 1; 1)-topos is: • The category of objects obtained by \homotopically gluing together" copies of some collection of \model objects" in specified ways. • The free cocompletion of a small (1; 1)-category preserving certain well-behaved colimits. • An accessible left exact localization of an (1; 1)-category of presheaves. They are a powerful tool for studying all kinds of \geometry" (topological, algebraic, differential, cohesive, etc.). It has long been expected that (1; 1)-toposes are models of HoTT, but coherence problems have proven difficult to overcome. Main Theorem Theorem (S.) Every (1; 1)-topos can be given the structure of a model of \Book" HoTT with strict univalent universes, closed under Σs, Πs, coproducts, and identity types. Caveats for experts: 1 Classical metatheory: ZFC with inaccessible cardinals. 2 We assume the initiality principle. 3 Only an interpretation, not an equivalence. 4 HITs also exist, but remains to show universes are closed under them. Towards killer apps Example 1 Hou{Finster{Licata{Lumsdaine formalized a proof of the Blakers{Massey theorem in HoTT. 2 Later, Rezk and Anel{Biedermann{Finster{Joyal unwound this manually into a new( 1; 1)-topos-theoretic proof, with a generalization applicable to Goodwillie calculus. 3 We can now say that the HFLL proof already implies the (1; 1)-topos-theoretic result, without manual translation. (Modulo closure under HITs.) Outline 1 Type-theoretic model toposes 2 Left exact localizations 3 Injective model structures 4 Remarks Review of model-categorical semantics We can interpret type theory in a well-behaved model category E : Type theory Model category Type Γ ` A Fibration ΓA Γ Term Γ ` a : A Section Γ ! ΓA over Γ Id-type Path object . -
Locally Cartesian Closed Categories, Coalgebras, and Containers
U.U.D.M. Project Report 2013:5 Locally cartesian closed categories, coalgebras, and containers Tilo Wiklund Examensarbete i matematik, 15 hp Handledare: Erik Palmgren, Stockholms universitet Examinator: Vera Koponen Mars 2013 Department of Mathematics Uppsala University Contents 1 Algebras and Coalgebras 1 1.1 Morphisms .................................... 2 1.2 Initial and terminal structures ........................... 4 1.3 Functoriality .................................... 6 1.4 (Co)recursion ................................... 7 1.5 Building final coalgebras ............................. 9 2 Bundles 13 2.1 Sums and products ................................ 14 2.2 Exponentials, fibre-wise ............................. 18 2.3 Bundles, fibre-wise ................................ 19 2.4 Lifting functors .................................. 21 2.5 A choice theorem ................................. 22 3 Enriching bundles 25 3.1 Enriched categories ................................ 26 3.2 Underlying categories ............................... 29 3.3 Enriched functors ................................. 31 3.4 Convenient strengths ............................... 33 3.5 Natural transformations .............................. 41 4 Containers 45 4.1 Container functors ................................ 45 4.2 Natural transformations .............................. 47 4.3 Strengths, revisited ................................ 50 4.4 Using shapes ................................... 53 4.5 Final remarks ................................... 56 i Introduction