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95
Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 36 (7 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
Homotopy colimits  comparison lemmas for combinatorial applications
, 1997
"... We provide a "toolkit " of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used on quite different fields of applications. We demonstrate this with respect to 1. Bjorner's " ..."
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Cited by 24 (2 self)
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We provide a &quot;toolkit &quot; of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used on quite different fields of applications. We demonstrate this with respect to 1. Bjorner's "Generalized Homotopy Complementation Formula" [4], 2. the topology of toric varieties, 3. the study of homotopy types of arrangements of subspaces, 4. the analysis of homotopy types of subgroup complexes.
Homotopytheoretic aspects of 2–monads
 J. Homotopy Relat. Struct
"... We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the e ..."
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Cited by 19 (2 self)
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We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2categories, including a model structure on the 2category of algebras for a 2monad T, and a model structure on a 2category of 2monads on a fixed 2category K. 1
OPERADS, ALGEBRAS, MODULES, AND MOTIVES
"... Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and ..."
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Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and conversion theorems III Derived categories from a topological point of view IV Rational derived categories and mixed Tate motives V Derived categories of modules over E ∞ algebras In differential algebra, operads are systems of parameter chain complexes for multiplication on various types of differential graded algebras “up to homotopy”, for example commutative algebras, nLie algebras, nbraid algebras, etc. Our primary focus is the development of the concomitant theory of modules up to homotopy and the study of both classical derived categories of modules over DGA’s and derived categories of modules up to homotopy over DGA’s up to homotopy. Examples of such derived categories provide the appropriate setting for one approach to mixed Tate motives in algebraic geometry, both rational and integral.
Stability for holomorphic spheres and Morse theory
 in Geometry and topology: Aarhus
, 1998
"... Abstract. In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (X, ω) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operations, the space of based holomorp ..."
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Abstract. In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (X, ω) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operations, the space of based holomorphic maps from a sphere to X, becomes homotopy equivalent to the space of all continuous maps, lim − → Holx0 (P1, X) ≃ Ω 2 X. This limit will be viewed as a kind of stabilization of Holx0 (P1, X). We conjecture that this stability property holds if and only if an evaluation map E: lim Holx0 (P1, X) → X is a quasifibration. In this paper we will prove that in the presence of this quasifibration condition, then the stability property holds if and only if the Morse theoretic flow category (defined in [4]) of the symplectic action functional on the Z cover of the loop space, ˜ LX, defined by the symplectic form, has a classifying space that realizes the homotopy type of ˜ LX. We conjecture that in the presence of this quasifibration condition, this Morse theoretic condition always holds. We will prove this in the case of X a homogeneous space, thereby giving an alternate proof of the stability theorem for holomorphic spheres for a projective homogeneous variety originally due to Gravesen [7].
SEMITOPOLOGICAL KTHEORY USING FUNCTION COMPLEXES
"... The semitopological Ktheory Ksemi ∗ (X) of a quasiprojective complex algebraic variety X is based on the notion of algebraic vector bundles modulo algebraic equivalence. This theory is given as the homotopy groups of an infinite loop space Ksemi (X) which is equipped with maps Kalg (X) → Ksemi ( ..."
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Cited by 15 (6 self)
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The semitopological Ktheory Ksemi ∗ (X) of a quasiprojective complex algebraic variety X is based on the notion of algebraic vector bundles modulo algebraic equivalence. This theory is given as the homotopy groups of an infinite loop space Ksemi (X) which is equipped with maps Kalg (X) → Ksemi (X), Ksemi (X) → Ktop(Xan) whose composition is the natural map from the algebraic Ktheory of X to the topological Ktheory of the underlying analytic space X an of X. We give an explicit description of Ksemi 0 (X) in terms of K0(X), a description of Ksemi q (−) in terms of Ksemi 0 (−) for projective varieties, a Poincaré duality theorem for projective varieties, and a computation of Ksemi (X) whenever X is a product of projective spaces or a smooth complete curve. For X a smooth quasiprojective variety, there are natural Chern class maps from K semi ∗ (X) to morphic cohomology compatible with similarly defined Chern class maps from algebraic Ktheory to motivic cohomology and compatible with the classical Chern class maps from topological Ktheory to the singular cohomology of Xan.
Yang–Mills theory over surfaces and the AtiyahSegal theorem
, 2008
"... Abstract. In this paper we explain how Morse theory for the YangMills functional can be used to prove an analogue, for surface groups, of the AtiyahSegal theorem. Classically, the AtiyahSegal theorem relates the representation ring R(Γ) of a compact group Γ to the complex Ktheory of the classify ..."
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Cited by 10 (7 self)
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Abstract. In this paper we explain how Morse theory for the YangMills functional can be used to prove an analogue, for surface groups, of the AtiyahSegal theorem. Classically, the AtiyahSegal theorem relates the representation ring R(Γ) of a compact group Γ to the complex Ktheory of the classifying space BΓ. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson’s deformation Ktheory spectrum Kdef(Γ) (the homotopytheoretical analogue of R(Γ)). Our main theorem provides an isomorphism in homotopy K ∗ def (π1Σ) ∼ = K ∗ (Σ) for all compact, aspherical surfaces Σ and all ∗> 0. Combining this result with work of Lawson, we obtain homotopy theoretical information about the stable moduli space of flat connections over surfaces. 1.
Homotopy algebras for operads
"... We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically ..."
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We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy Palgebra in M is, provided only that some of the morphisms in M have been marked out as ‘homotopy equivalences’. The bulk of the paper consists of examples of homotopy algebras. We show that any loop space is a homotopy monoid, and, in fact, that any nfold loop space is an nfold homotopy monoid in an appropriate sense. We try to compare weakened algebraic structures such as A∞spaces, A∞algebras and nonstrict monoidal categories to our homotopy algebras, with varying degrees of success. We also prove results on ‘change of base’, e.g. that the classifying space of a homotopy monoidal category is a homotopy topological monoid. Finally, we
Geometric cobordism categories
 ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Stanford University. MR2713365
"... In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principle bundles, or complex structures along with a holomorphic map to a target complex manifold. ..."
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In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principle bundles, or complex structures along with a holomorphic map to a target complex manifold. A general notion of “geometric structure” is defined using sheaf theoretic constructions. Our main theorem is the identification of the homotopy type of such cobordism categories in terms certain Thom spectra. This extends work of GalatiusMadsenTillmannWeiss who identify the homotopy type of cobordism categories of manifolds with fiberwise structures on their tangent bundles.