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## ‣ Stringy K-theory and the Chern character

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Geometry#Mathematics - Differential Geometry#Mathematics - K-Theory and Homology#Mathematics - Quantum Algebra#19L47#53D45#55N15#14N35#55R65#57R20

For a finite group G acting on a smooth projective variety X, we construct
two new G-equivariant rings: first the stringy K-theory of X, and second the
stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct
a new ring called the full orbifold K-theory of Y. For a global quotient
Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra
of the full orbifold K-theory of the the stack Y and is linearly isomorphic to
the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a
different, ``quantum,'' product, which respects the natural group grading. We
prove there is a ring isomorphism, the stringy Chern character, from stringy
K-theory to stringy cohomology, and a ring homomorphism from full orbifold
K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy
Grothendieck-Riemann-Roch for etale maps.
We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's
construction. Since our constructions do not use complex curves, stable maps,
admissible covers, or moduli spaces, our results simplify the definitions of
Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of
Abramovich-Graber-Vistoli's orbifold Chow.
We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler
Resolution Conjecture holds for symmetric products.
Our results hold both in the algebro-geometric category and in the
topological category for equivariant almost complex manifolds.; Comment: Exposition improved and additional details provided. To appear in
Inventiones Mathematicae

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## ‣ Gromov-Witten theory and Brane actions I : categorification and K-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/05/2015
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#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - K-Theory and Homology#14N35, 53N45, 19, 55U40, 18G55

This is a draft version. Let X be a smooth projective variety. Using the idea
of brane actions discovered by To\"en, we construct a lax associative action of
the operad of stable curves of genus zero on the variety X seen as an object in
correspondences in derived stacks. This action encodes the Gromov-Witten theory
of X in purely geometrical terms and induces an action on the derived category
Qcoh(X) which allows us to recover the formulas for Quantum K-theory of
Givental-Lee. This paper is the first step of a larger project. We believe that
this action in correspondences encodes the full classical cohomological
Gromov-Witten invariants of X. This will appear in a second paper.; Comment: preliminary version, 57 pages

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## ‣ Non-connective K-theory of relative exact categories

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/03/2013
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#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - K-Theory and Homology

The main objective of this paper is to propose a definition of non-connective
K-theory for a wide class of relative exact categories which, in general, do
not satisfy the factorization axiom and confirm that it agrees with the
non-connective K-theory for exact categories and complicial exact categories
with weak equivalences. The main application is to study the topological
filtrations of non-connective K-theory of a noetherian commutative ring with
unit in terms of Koszul cubes.

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## ‣ Parametrized K-Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/04/2013
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#Mathematics - K-Theory and Homology#Mathematics - Commutative Algebra#Mathematics - Algebraic Geometry#Mathematics - Category Theory#18F25, 19D99 (Primary) 13D15, 14F05, 14F20, 18D10, 18D30, 18D99,
18E10, 18F10, 19E08 (Secondary)

In nature, one observes that a K-theory of an object is defined in two steps.
First a "structured" category is associated to the object. Second, a K-theory
machine is applied to the latter category to produce an infinite loop space. We
develop a general framework that deals with the first step of this process. The
K-theory of an object is defined via a category of "locally trivial" objects
with respect to a pretopology. We study conditions ensuring an exact structure
on such categories. We also consider morphisms in K-theory that such contexts
naturally provide. We end by defining various K-theories of schemes and
morphisms between them.; Comment: 31 pages

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## ‣ Twisted and untwisted K-theory quantization, and symplectic topology

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/08/2015
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#Mathematics - Symplectic Geometry#Mathematical Physics#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - K-Theory and Homology

A prequantization space $(P,\alpha)$ is a principal $S^1$-bundle with a
connection one-form $\alpha$ over a symplectic manifold $(M,\omega),$ with
curvature given by the symplectic form. In particular $\alpha$ is a contact
form. Using the theory of $Spin ^{c} $ Dirac quantization, we set up natural
K-theory invariants of structure group $Cont _{0} (P, \alpha) $ fibrations of
prequantization spaces. We further construct twisted K-theory invariants of
Hamiltonian $(M, \omega)$ fibrations. As an application we prove that the
natural map $BU(r) \to BU$ of classifying spaces factors as $BU(r) \to B
\mathcal{Q}(r \to BU,$ where $\mathcal{Q}(r)=Cont_0(S^{2r-1},\alpha_{std})$ for
the standard contact form on the odd-dimensional sphere. As a corollary we show
that the natural map $BU(r) \rightarrow B\mathcal{Q}(r)$ induces a surjection
on complex K-theory and on integral cohomology, strengthening a theorem of
Spacil \cite{SpacilThesis,CasalsSpacil} for rational cohomology, and that it
induces an injection on integral homology. Furthermore, we improve a theorem of
Reznikov, showing the injectivity on homotopy groups of the natural map $BU (r)
\to B \text{Ham} (\mathbb{CP} ^{r-1}, \omega )$, in the stable range. Finally,
we produce examples of non-trivial $\mathcal{Q}(r)$ and $\text{Ham}
(\mathbb{CP} ^{r-1}...

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## ‣ The ring structure for equivariant twisted K-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - K-Theory and Homology#High Energy Physics - Theory#Mathematical Physics#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - Operator Algebras#19L47 (Primary)#55N91, 46L80, 20L05 (Secondary)

We prove, under some mild conditions, that the equivariant twisted K-theory
group of a crossed module admits a ring structure if the twisting 2-cocycle is
2-multiplicative. We also give an explicit construction of the transgression
map $T_1: H^*(\Gamma;A) \to H^{*-1}((N\rtimes \Gamma;A)$ for any crossed module
$N\to \Gamma$ and prove that any element in the image is
$\infty$-multiplicative. As a consequence, we prove that, under some mild
conditions, for a crossed module $N \to \gm$ and any $e \in
\check{Z}^3(\Gamma;S^1)$, that the equivariant twisted K-theory group
$K^*_{e,\Gamma}(N)$ admits a ring structure. As an application, we prove that
for a compact, connected and simply connected Lie group G, the equivariant
twisted K-theory group $K_{[c], G}^* (G)$ is endowed with a canonical ring
structure $K^{i+d}_{[c],G}(G)\otimes K^{j+d}_{[c],G}(G)\to K^{i+j+d}_{[c],
G}(G)$, where $d=dim G$ and $[c]\in H^2(G\rtimes G;S^1)$.; Comment: 47 pages. To appear in Crelle

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## ‣ Algebraic K-theory is stable and admits a multiplicative structure for module objects

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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After recognizing higher homotopy coherences, algebraic K-theory can be
regarded as a functor from stable $\infty$-categories to $\infty$-categories.
We establish the stability theoremm which states that the algebraic K-theory of
a stable $\infty$-category is actually a stable $\infty$-category itself. This
is a generalization of the statement that algebraic K-theory is a functor from
spectra to spectra. We then prove a result which provides a simpler
interpretation of the algebraic K-theory of ring spectra. In order to do this,
we compute the algebraic K-theory of an $\infty$-category of modules, and
establish that it is an $\infty$-category of modules itself. This result, known
as the multiplicativity theorem, vastly generalizes results obtained by
Elmendorf and Mandell. Since the algebraic K-theory of a ring spectrum $R$ is
the algebraic K-theory of the $\infty$-category of perfect modules over $R$,
this provides a simpler interpretation of the algebraic K-theory of ring
spectra. Using this result, we prove an $\infty$-categorical counterpart of the
derived Morita context for flat rings, which shows that algebraic K-theory is a
homotopy coherent version of Morita theory.; Comment: 12 pages. Any updates to this version of the paper will be available
from the author's webpage. Comments are welcome!

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## ‣ Connective algebraic K-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/12/2012
Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#14C25, 19E15 (Primary) 19E08 14F42, 55P42 (Secondary)

We examine the theory of connective algebraic K-theory, CK, defined by taking
the -1 connective cover of algebraic K-theory with respect to Voevodsky's slice
tower in the motivic stable homotopy category. We extend CK to a bi-graded
oriented duality theory (CK', CK) in case the base scheme is the spectrum of a
field k of characteristic zero. The homology theory CK' may be viewed as
connective algebraic G-theory. We identify CK' theory in bi-degree (2n, n) on
some finite type k-scheme X with the image of K_0(M(X,n)) in K_0(M(X, n+1)),
where M(X,n) is the abelian category of coherent sheaves on X with support in
dimension at most n; this agrees with the (2n,n) part of the theory defined by
Cai. We also show that the classifying map from algebraic cobordism identifies
CK' with the universal oriented Borel-Morel homology theory
\Omega_*^{CK}:=\Omega_*\otimes_L\Z[\beta] having formal group law u+v-\beta uv
with coefficient ring \Z[\beta]. As an application, we show that every pure
dimension d finite type k scheme has a well-defined fundamental class [X] in
\Omega_d^{CK}(X), and this fundamental class is functorial with respect to
pull-back for lci morphisms. Finally, the fundamental class maps to the
fundamental class in G-theory after inverting \beta...

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## ‣ The Eta-invariant and Pontryagin duality in K-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - K-Theory and Homology#Mathematics - Analysis of PDEs#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - Operator Algebras#Mathematics - Spectral Theory#58J28, 19L64 (Primary)#58J22, 19K56, 58J40 (Secondary)

The topological significance of the spectral Atiyah-Patodi-Singer
eta-invariant is investigated under the parity conditions of P. Gilkey. We show
that twice the fractional part of the invariant is computed by the linking
pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin
duality implies the nondegeneracy of the linking form. An example of a
nontrivial fractional part for an even-order operator is presented. This result
answers the question of P. Gilkey (1989) concerning the existence of even-order
operators on odd-dimensional manifolds with nontrivial fractional part of
eta-invariant.; Comment: 24 pages, 1 figure; final version; see
http://www.kluweronline.com/issn/0001-4346/contents

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## ‣ Algebraic K-theory of strict ring spectra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/03/2014
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#Mathematics - Algebraic Topology#Mathematics - Geometric Topology#Mathematics - K-Theory and Homology#Mathematics - Number Theory#19D10, 55P43, 19F27, 57R50

We view strict ring spectra as generalized rings. The study of their
algebraic K-theory is motivated by its applications to the automorphism groups
of compact manifolds. Partial calculations of algebraic K-theory for the sphere
spectrum are available at regular primes, but we seek more conceptual answers
in terms of localization and descent properties. Calculations for ring spectra
related to topological K-theory suggest the existence of a motivic cohomology
theory for strictly commutative ring spectra, and we present evidence for
arithmetic duality in this theory. To tie motivic cohomology to Galois
cohomology we wish to spectrally realize ramified extensions, which is only
possible after mild forms of localization. One such mild localization is
provided by the theory of logarithmic ring spectra, and we outline recent
developments in this area.; Comment: Contribution to the proceedings of the ICM 2014 in Seoul

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## ‣ A Theory of Adjoint Functors--with some Thoughts about their Philosophical Significance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/11/2005
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The question "What is category theory" is approached by focusing on universal
mapping properties and adjoint functors. Category theory organizes mathematics
using morphisms that transmit structure and determination. Structures of
mathematical interest are usually characterized by some universal mapping
property so the general thesis is that category theory is about determination
through universals. In recent decades, the notion of adjoint functors has moved
to center-stage as category theory's primary tool to characterize what is
important and universal in mathematics. Hence our focus here is to present a
theory of adjoint functors, a theory which shows that all adjunctions arise
from the birepresentations of "chimeras" or "heteromorphisms" between the
objects of different categories. Since representations provide universal
mapping properties, this theory places adjoints within the framework of
determination through universals. The conclusion considers some unreasonably
effective analogies between these mathematical concepts and some central
philosophical themes.; Comment: 58 pages. Forthcoming in: What is Category Theory? Giandomenico Sica
ed., Milan: Polimetrica

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## ‣ On determinant functors and $K$-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - Number Theory#19A99, 19B99, 18F25, 18G50, 18G55, 18E10, 18E30

In this paper we introduce a new approach to determinant functors which
allows us to extend Deligne's determinant functors for exact categories to
Waldhausen categories, (strongly) triangulated categories, and derivators. We
construct universal determinant functors in all cases by original methods which
are interesting even for the known cases. Moreover, we show that the target of
each universal determinant functor computes the corresponding $K$-theory in
dimensions 0 and 1. As applications, we answer open questions by Maltsiniotis
and Neeman on the $K$-theory of (strongly) triangulated categories and a
question of Grothendieck to Knudsen on determinant functors. We also prove
additivity theorems for low-dimensional $K$-theory and obtain generators and
(some) relations for various $K_{1}$-groups.; Comment: 73 pages. We have deeply revised the paper to make it more
accessible, it contains now explicit examples and computations. We have
removed the part on localization, it was correct but we didn't want to make
the paper longer and we thought this part was the less interesting one.
Nevertheless it will remain here in the arXiv, in version 1. If you need it
in your research, please let us know

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## ‣ The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Number Theory

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible
in its ring of regular functions, an ample family of line bundles, and a global
bound on the virtual mod-2 cohomological dimensions of its residue fields. We
prove that the comparison map from the hermitian K-theory of X to the homotopy
fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence
in general, and an integral equivalence when X has no formally real residue
field. We also show that the comparison map between the higher
Grothendieck-Witt (hermitian K-) theory of X and its \'etale version is an
isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum
conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of
complex algebraic varieties and rings of 2-integers in number fields, and hence
values of Dedekind zeta-functions.; Comment: 17 pages, to appear in Adv. Math

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## ‣ K-Theory in Quantum Field Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/06/2002
Português

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#Mathematical Physics#High Energy Physics - Theory#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - K-Theory and Homology#81T30, 81T45, 81T50, 19L99

We survey three different ways in which K-theory in all its forms enters
quantum field theory. In Part 1 we give a general argument which relates
topological field theory in codimension two with twisted K-theory, and we
illustrate with some finite models. Part 2 is a review of pfaffians of Dirac
operators, anomalies, and the relationship to differential K-theory. Part 3 is
a geometric exposition of Dirac charge quantization, which in superstring
theories also involves differential K-theory. Parts 2 and 3 are related by the
Green-Schwarz anomaly cancellation mechanism. An appendix, joint with Jerry
Jenquin, treats the partition function of Rarita-Schwinger fields.; Comment: 56 pages, expanded version of lectures at "Current Developments in
Mathematics"

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## ‣ Comparison between algebraic and topological K-theory of locally convex algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - K-Theory and Homology#Mathematics - Rings and Algebras#18G, 19K, 46H, 46L80, 46M, 58B34

This paper is concerned with the algebraic K-theory of locally convex
algebras stabilized by operator ideals, and its comparison with topological
K-theory. We show that the obstruction for the comparison map between algebraic
and topological K-theory to be an isomorphism is (absolute) algebraic cyclic
homology and prove the existence of an 6-term exact sequence.
We show that cyclic homology vanishes in the case when J is the ideal of
compact operators and L is a Frechet algebra with bounded app. unit. This
proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki
and announced in his paper "Algebraic K-theory and functional analysis", First
European Congress of Mathematics, Vol. II (Paris, 1992), 485--496, Progr.
Math., 120, Birkh\"auser, Basel, 1994.
We also consider stabilization with respect to a wider class of operator
ideals, called sub-harmonic. We study the algebraic K-theory of the tensor
product of a sub-harmonic ideal with an arbitrary complex algebra and prove
that the obstruction for the periodicity of algebraic K-theory is again cyclic
homology.
The main technical tools we use are the diffeotopy invariance theorem of
Cuntz and the second author (which we generalize), and the excision theorem for
infinitesimal K-theory...

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## ‣ Homotopy invariance of higher K-theory for abelian categories

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Geometry#Mathematics - Commutative Algebra#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - K-Theory and Homology

The main theorem in this paper is that the base change functor from a
noetherian abelian category to its noetherian polynomial category induces an
isomorphism on K-theory. The main theorem implies the well-known fact that
A^1-homotopy invariance of K'-theory for noetherian schemes.; Comment: arXiv admin note: substantial text overlap with arXiv:1104.4240

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## ‣ Unified Foundations for Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/03/2004
Português

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There are different meanings of foundation of mathematics: philosophical,
logical, and mathematical. Here foundations are considered as a theory that
provides means (concepts, structures, methods etc.) for the development of
whole mathematics. Set theory has been for a long time the most popular
foundation. However, it was not been able to win completely over its rivals:
logic, the theory of algorithms, and theory of categories. Moreover, practical
applications of mathematics and its inner problems caused creation of different
generalization of sets: multisets, fuzzy sets, rough sets etc. Thus, we
encounter a problem: Is it possible to find the most fundamental structure in
mathematics? The situation is similar to the quest of physics for the most
fundamental "brick" of nature and for a grand unified theory of nature. It is
demonstrated that in contrast to physics, which is still in search for a
unified theory, in mathematics such a theory exists. It is the theory of named
sets.

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## ‣ Almost ring theory - sixth release

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Geometry#Mathematics - Commutative Algebra#Mathematics - Number Theory#Mathematics - Rings and Algebras#13D03, 12J20, 14A99, 14F99, 18D10, 14G22

We develop almost ring theory, which is a domain of mathematics somewhere
halfway between ring theory and category theory (whence the difficulty of
finding appropriate MSC-class numbers). We apply this theory to valuation
theory and to p-adic analytic geometry. You should really have a look at the
introductions (each chapter has one).; Comment: This is the sixth - and assuredly final - release of "Almost ring
theory". It is about 230 page long; it is written in AMSLaTeX and uses XYPic
and a few not so standard fonts. Any future corrections (mainly typos, I
expect) will be found on my personal web page:
http://www.math.u-bordeaux.fr/~ramero/

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## ‣ Homotopy Type Theory: Univalent Foundations of Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/08/2013
Português

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#Mathematics - Logic#Computer Science - Programming Languages#Mathematics - Algebraic Topology#Mathematics - Category Theory

Homotopy type theory is a new branch of mathematics, based on a recently
discovered connection between homotopy theory and type theory, which brings new
ideas into the very foundation of mathematics. On the one hand, Voevodsky's
subtle and beautiful "univalence axiom" implies that isomorphic structures can
be identified. On the other hand, "higher inductive types" provide direct,
logical descriptions of some of the basic spaces and constructions of homotopy
theory. Both are impossible to capture directly in classical set-theoretic
foundations, but when combined in homotopy type theory, they permit an entirely
new kind of "logic of homotopy types". This suggests a new conception of
foundations of mathematics, with intrinsic homotopical content, an "invariant"
conception of the objects of mathematics -- and convenient machine
implementations, which can serve as a practical aid to the working
mathematician. This book is intended as a first systematic exposition of the
basics of the resulting "Univalent Foundations" program, and a collection of
examples of this new style of reasoning -- but without requiring the reader to
know or learn any formal logic, or to use any computer proof assistant.; Comment: 465 pages. arXiv v1: first-edition-257-g5561b73...

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## ‣ Categorical Foundations for K-Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/11/2011
Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Category Theory#19-02, 18F25 (Primary) 18D05, 18D10, 18D30, 18D35, 18F10, 19E08,
19L99, 55N15 (Secondary)

Recall that the definition of the $K$-theory of an object C (e.g., a ring or
a space) has the following pattern. One first associates to the object C a
category A_C that has a suitable structure (exact, Waldhausen, symmetric
monoidal, ...). One then applies to the category A_C a "$K$-theory machine",
which provides an infinite loop space that is the $K$-theory K(C) of the object
C.
We study the first step of this process. What are the kinds of objects to be
studied via $K$-theory? Given these types of objects, what structured
categories should one associate to an object to obtain $K$-theoretic
information about it? And how should the morphisms of these objects interact
with this correspondence?
We propose a unified, conceptual framework for a number of important examples
of objects studied in $K$-theory. The structured categories associated to an
object C are typically categories of modules in a monoidal (op-)fibred
category. The modules considered are "locally trivial" with respect to a given
class of trivial modules and a given Grothendieck topology on the object C's
category.; Comment: 176 + xi pages. This monograph is a revised and augmented version of
my PhD thesis. The official thesis is available at
http://library.epfl.ch/en/theses/?nr=4861

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