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Markl M., Shnider S., Stasheff J. — Operads in Algebra, Topology and Physics
Markl M., Shnider S., Stasheff J. — Operads in Algebra, Topology and Physics

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Название: Operads in Algebra, Topology and Physics

Авторы: Markl M., Shnider S., Stasheff J.

Аннотация:

Operads are mathematical devices which describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of "homotopy" where they play a key role in organizing hierarchies of higher homotopies. Significant examples first appeared in the sixties though the formal definition and appropriate generality waited for the seventies. These early occurrences were in algebraic topology in the study of (iterated) loop spaces and their chain algebras. In the nineties, there was a renaissance and further development of the theory inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, not least, theoretical physics, especially string field theory and deformation quantization. The generalization of quadratic duality (e.g., Lie algebras as dual to commutative algebras) together with the property of Koszulness in an essentially operadic context provided an additional computational tool for studying homotopy properties outside of the topological setting. The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature. There is a chapter on topology, reviewing classical results with the emphasis on the $W$-construction and homotopy invariance. Another chapter describes the (co)homology of operad algebras, minimal models, and homotopy algebras. A chapter on geometry focuses on the configuration spaces and their compactifications. A final chapter deals with cyclic and modular operads and applications to graph complexes and moduli spaces of surfaces of arbitrary genus.


Язык: en

Рубрика: Computer science/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 178

Добавлена в каталог: 12.04.2014

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Предметный указатель
$*_{g,n}$ modular n-crolla of genus g      271
$*_{n}$ n-star      250
$A(\infty)$-algebra      195
$A_{\infty}$-algebra      13 172 195
$A_{\infty}$-category      14
$A_{\infty}$-map      113
$A_{\infty}$-space      10
$B_{\infty}$ Baues operad      30
$B_{\infty}$-algebra      30
$CH^{*}(A;A)$ Hochschild cochains of A with coefficients in itself      29 104
$CH_{*}(A;A)$ Hochschild chains of A with coefficients in itself      259
$Col_{K}$ category of K-collections      85
$Con(\mathbb{R}^{k},n)$ configuration space of n distinct labeled points in $\mathbb{R}^{k}$      26 100
$Co\mathcal{E}nd_{X}$ coendomorphism operad      43
$C^{*}_{\mathcal{P}}(V;V)$ $\mathcal{P}$-algebra cochain complex with coefficients in itself      177 339
$C^{-*}_{\mathcal{P}}(V)$ $\mathcal{P}$-algebra chain complex with trivial coefficients      173
$C_{\infty}$-algebra      19 172 196
$dg\psi Op$ category of differential graded pseudo-operads      126
$D\mathcal{P}$ $\sigma$-module of decomposables of an operad $\mathcal{P}$      187
$D^{k}$, $B^{k}$ standard unit k-dimensional disk (ball)      204
$edge(\Gamma)$ set of edges of a graph $\Gamma$      269
$E^{\#}$ linear dual of a $\Sigma$-module      141
$E^{\vee}$ Czech dual of a $\Sigma$-module E      142
$e_{2}$ operad for Gerstenhaber algebras      30 176
$E_{\infty}$-operad      12
$Flag(\Gamma)$ set of flags (half-edges) of a graph $\Gamma$      269
$F_{k}$ simplified notation for the operad $F_{\mathbb{R}^{k}}$      27 226
$F_{k}(n)$ simplified notation for the space $F_{k}(n)$      27 225
$F_{V}$ operad $\{F_{V}(n)\}_{n\geq1}$      225
$F_{V}(n)$ compactification of the moduli space $\r{F}_{V}(n)$      225
$F_{\mathfrak{D}}(A)$ Feynman transform of a $\mathfrak{D}$-modular operad A      287
$g(\Gamma)$ genus of a labeled graph $\Gamma$      270
$GC^{(n)}_{*}$ graph complex      301
$Gr^{(n)}$ set of connected graphs with Euler characteristic 1-n      301
$Gr^{met}_{(n)}$ set of graphs $\Gamma\in Gr^{(n)}$ with a metric      301
$G_{\infty}$ minimal model of the operad $e_{2}$      30
$Harr_{*}(A, M)$ Harrison homology of A with coefficients in M      266
$HH_{*}(A;A)$ Hochschild homology of A with coefficients in itself      259
$H^{*}_{DR}(A)$ de Rham cohomology of an algebra A      260
$H^{*}_{\mathcal{P}}(A)(V;V)$ operadic cohomology of a $\mathcal{P}$-algebra V with coefficients in itself      177
$H^{\lambda}_{*}(A)$ cyclic homology of an associative algebra A      259
$H_{n}$-space      104
$IFlag(\Gamma)$ set of internal flags of a graph $\Gamma$      285
$Iso(\mathcal{D})$ category of isomorphisms of a category $\mathcal{D}$      72 271
$K_{n}$ Stasheff's associahedron      9 56
$L(\infty)$-algebra      172 195
$Leg(\Gamma)$ set of legs (half-edges) of a graph $\Gamma$      269
$L_{\infty}$-algebra      17 195
$MC_{g,s}$ mapping class group of a genus g surface with s distinct labeled points      292
$P_{n}$ permutahedron      97
$Q\mathcal{P}$ $\Sigma$-module of indecomposables of an operad $\mathcal{P}$      188
$RGC^{g,s}_{*}$ ribbon graph complex      295
$RGr^{met}_{g,s}$ set of ribbon graphs $\Gamma\in RGr_{g,s}$ with a metric      292
$RGr_{g,s}$ set of ribbon graphs $\Gamma$ such that $Surf(\Gamma)$ has s holes and genus g      292
$Set_{f}$ category of finite sets      40
$Set_{f}-Mod$ category of $Set_{f}$-modules      40
$sgn_{n}$ signum representation of the symmetric group $\Sigma_{n}$      264
$SS^{p}_{n}$ set of surjection sequences      152
$Surf(\Gamma)$ surface associated to a ribbon graph $\Gamma$      292
$Surj_{f}$ category of surjections of finite sets      60
$Tree^{+}_{n}$ category of unrooted trees with legs labeled by {0,...,n}      250
$Tree_{n}$ category of labeled rooted n-trees      52
$Tree_{X}$ category of X-labeled rooted trees      52
$T_{[f]}$ tree corresponding to a surjection f      151
$U\mathcal{A}ss$ operad for associative algebras with unit      267
$U\mathcal{C}om$ operad for commutative associative algebras with unit      267 316
$U\mathcal{P}oiss$ operad for Poisson algebras with unit      267
$Vert(\Gamma)$ set of vertices of a graph $\Gamma$      269
$V^{\#}=(V^{\#},d^{\#})$ linear dual of a dg complex      121
$W(\mathcsr{P})$ W-construction on an operad $\mathcal{P}$      28 111
$W\mathcal{P}$-homomorphism      116
$W_{n}$ cyclohedron      241
$\bigwedge$ determinant operad      128
$\breve{\mathfrak{D}}$ dual of a cocycle $\mathfrak{D}$      284
$\chi(\sigma)$ skew-symmetric Koszul sign      196
$\circ_{i}$ composition (Gerstenhaber) product      6
$\delta$-complex      125
$\downarrow U$ desuspension of a graded vector space U      314
$\epsilon(\sigma)$ Koszul sign      195
$\Gamma$-space      101
$\Gamma((g,S))$ category of stable S-labeled graphs of genus g      271
$\Gamma(-)$ free operad functor      82
$\Gamma(C)$ dual graph of a stable curve C      304
$\Gamma_{+}(-)$ free cyclic operad functor      251
$\Gamma_{K}(-)$ free K-operad functor      88
$\Lambda(\mathcal{P},-)$ universal bilinear form      261
$\mathbb{CP}^{1}$ complex projective line      207
$\mathbb{M}(-)$ triple for modular operads      271
$\mathbb{M}_{\mathfrak{D}}(-)$ triple for $\mathfrak{D}$-twisted modular operads      281
$\mathbf{C}(\mathcal{P})$ cobar complex of an operad $\mathcal{P}$      20 121 125
$\mathbf{D}(\mathcal{P})$ dual dg operad of an operad $\mathcal{P}$      20 121 127
$\mathbf{f}$-order      156
$\mathbf{K}(\mathcal{P}^{!})$ Koszul complex of an operad $\mathcal{P}$      121 145
$\mathbf{K}(\mathcal{P}^{!})^{\#}$ dual Koszul complex of an operad $\mathcal{P}$      146
$\mathbf{N}(\mathcal{P})$ categorial cobar complex of an operad $\mathcal{P}$      121 149
$\mathbf{r}(U)$ reflection of a graded vector space U      314
$\mathcal{A}ss$ operad for associative algebras      20 44 168
$\mathcal{A}ss_{\infty}$ operad for $A_{\infty}$-algebras      20
$\mathcal{B(P)}$ bar construction on an operad $\mathcal{P}$      20
$\mathcal{BV}$ operad for Batalin — Vilkovisky algebras      206 257
$\mathcal{B}race$ operad for brace algebras      31
$\mathcal{B}_{k}$ operad for k-braid algebras      27
$\mathcal{C}mon$ operad for commutative associative monoids      111
$\mathcal{C}om$ operad for commutative associative algebras      20 44 168
$\mathcal{C}om_{\infty}$ operad for $C_{\infty}$-algebras      20
$\mathcal{C}_{1}$ little intervals operad      95
$\mathcal{C}_{2}$ little squares operad      95
$\mathcal{C}_{k}$ little k-cubes operad      12 94 203
$\mathcal{D}_{k}$ little k-disks operad      12 203
$\mathcal{EZ}$ Eilenberg — Zilber operad      105
$\mathcal{E}nd_{X}$ endomorphism operad      6 43 254
$\mathcal{E}tree(n)$ set of of isomorphism classes of n-trees with a special vertex      234
$\mathcal{F_{P}}(-)$ free $\mathcal{P}$-algebra functor      47
$\mathcal{F}^{c}_{\mathcal{P}}(-)$ cofree nilpotent $\mathcal{P}$-coalgebra functor      166
$\mathcal{HG}$ McClure — Smith operad      30 31
$\mathcal{H}=(\mathcal{H},Q)$ BRST complex      24 208
$\mathcal{H}_{rel}=(\mathcal{H}_{rel},Q)$ relative BRST complex      26
$\mathcal{L}eib$ operad for Leibniz algebras      257
$\mathcal{L}i$ linear isometries operad      106 107
$\mathcal{L}ie$ operad for Lie algebras      20 50
$\mathcal{L}ie_{\infty}$ operad for $L_{\infty}$-algebras      18
$\mathcal{M}on$ operad for associative topological monoids      28 99
$\mathcal{M}_{0,n}$ moduli space of n distinct labeled points in $\mathbb{CP}^{1}$      212
$\mathcal{M}_{g,n}$ moduli space of genus g Riemann surfaces with n distinct labeled points      292 305
$\mathcal{N}(n)$ moduli space of Riemann spheres with n+1 marked decorated points      25
$\mathcal{P}$ = $\langle E;R\rangle$ presentation of an operad $\mathcal{P}$      139
$\mathcal{P}$-algebra      46
$\mathcal{P}$-algebra chain complex      173
$\mathcal{P}$-algebra cochain complex      177
$\mathcal{P}$-algebra cohomology      177
$\mathcal{P}$-algebra homology      174
$\mathcal{P}$-coalgebra      165
$\mathcal{P}$-degree      177
$\mathcal{P}$-differential      173
$\mathcal{P}$-homomorphism      112
$\mathcal{P}oiss$ operad for Poisson algebras      176 256 261
$\mathcal{P}\langle -\rangle$ free $\mathcal{P}$-module functor      139
$\mathcal{P}\times G$ semidirect product of an operad V with a group G      204
$\mathcal{P}^{!}$ quadratic dual of a quadratic operad $\mathcal{P}$      21
$\mathcal{P}^{+}$ augmentation ideal of an augmented operad $\mathcal{P}$      187
$\mathcal{P}^{+}$ pseudo-operad associated to an operad $\mathcal{P}$      110
$\mathcal{P}^{\bot}$ operad $(\mathfrak{s}\mathcal{P}^{!})_{#}$      262
$\mathcal{P}_{B\rightarrow W}$ colored operad for homomorphisms of $\mathcal{P}$-spaces      115
$\mathcal{R}mtree$ pseudo-operad of reduced rooted metric trees      110
$\mathcal{R}mtree(n)$ space of isomorphism classes of reduced rooted metric n-trees      109 221
$\mathcal{R}m\underline{tree}$ non-$\Sigma$ pseudo-operad of reduced rooted planar metric trees      112
$\mathcal{R}m\underline{tree}(n)$ space of isomorphism classes of reduced rooted metric planar n-trees      112
$\mathcal{R}tree$ pseudo-operad of reduced rooted trees      55
$\mathcal{R}tree(n)$ set of isomorphism classes of reduced rooted n-trees      55
$\mathcal{R}\underline{tree}$ non-$\Sigma$ operad of reduced planar rooted trees      55 109
$\mathcal{R}\underline{tree}(n)$ set of isomorphism classes of reduced planar rooted n-trees      55
$\mathcal{T}ree$ operad of rooted trees      8 54
$\mathcal{T}ree(n)$ set of isomorphism classes of rooted n-trees      8
$\mathcal{T}ree(X)$ set of isomorphism classes of X-labeled trees      53
$\mathcal{T}^{dec}_{g,s}$ decorated Teichmueller space of genus g surfaces with s distinct labeled points      291
$\mathcal{T}_{g,s}$ Teichmueller space of genus g surfaces with s distinct labeled points      291
$\mathfrak{D}$-twisted modular operad      286
$\mathfrak{f}Con(M,n)$ framed configuration space of n distinct labeled points in a manifold M      210 234
$\mathfrak{f}\mathcal{D}_{k}$ framed little k-disks operad      13 203
$\mathfrak{f}\overline{Con}(M,n)$ compactification of the space $\mathfrak{f}Con(M,n)$      234
$\mathfrak{M}^{dec}_{g,s}$ moduli space of genus g surfaces with s distinct unlabeled decorated points      292
$\mathfrak{M}_{g,s}$ moduli space of genus g surfaces with s distinct unlabeled points      292
$\mathfrak{s}A$ operadic suspension of a $\Sigma$-module A      127 258
$\mathfrak{s}\mathcal{E}$ modular suspension of a modular $\Sigma$-module $\mathcal{E}$      283
$\mathfrak{s}^{-1}A$ operadic desuspension of a $\Sigma$-module A      127
$\mathfrak{Z}(h)$ zero set of a metric $h\in Met(T)$      221
$\mathtt{Mod}(\mathcal{P})$ modular completion of an operad $\mathcal{P}$      317
$\mathtt{Mod}_{\mathbf{k}}$ category of $\mathbf{k}$-modules      38
$\mathtt{Mod}_{\mathcal{P}}$ category of $\mathcal{P}$-modules      139
$\mathtt{Mon}(\mathcal{C})$ category of $\mathcal{C}$-monoids      40
$\mathtt{Nerve}_{\bullet}(\mathcal{C})$ nerve of a category $\mathcal{C}$      272
$\Omega^{*}_{A}$ module of differentials of an algebra A      260
$\Omega^{*}_{DR}(M)$ dg algebra of de Rham forms      29
$\Omega^{k}X$ k-fold loop space of a topological space X      93
$\otimes$-product of operads      118
$\overline{Con}(M,n)$ compactification of the configuration space Con(M,n)      234 239
$\overline{Vert}(T)$ set of all (including external) vertices of a tree T      51
$\overline{\mathcal{M}}$ configuration pseudo-operad $\{\overline{\mathcal{M}}(n)\}_{n\geq2}$      25 216
$\overline{\mathcal{M}}$ modular operad $\{\overline{\mathcal{M}}(g,n)\}_{(g,n)\in \mathfrak{G}}$      306
$\overline{\mathcal{M}}(g,n)$,$\overline{\mathcal{M}}_{g,n+i}$ compactification of the moduli space $\mathcal{M}_{g,n+i}$      304
$\overline{\mathcal{M}}(n)$ compactification of the moduli space $\mathcal{M}_{0,n+1}$      213 305
$\Psi Op$ category of pseudo-operads      67
$\Psi(-)$ free pseudo-operad functor      81
$\Psi_{+}(-)$ free cyclic pseudo-operad functor      251
$\r{F}_{k}(n)$ simplified notation for the space $\r{F}_{\mathbb{R}^{k}}$      220
$\r{F}_{k}(n)$ simplified notation for the space $\r{F}_{\mathbb{R}^{k}}(n)$      220
$\r{F}_{V}$ $\Sigma$-module $\{\r{F}_{V}(n)\}_{n\geq1}$      220
$\r{F}_{V}(n)$ moduli space $\r{F}_{V}(n):=Con(V,n)/Aff(V)$      220
$\Sigma$ symmetric groupoid      40
$\Sigma$-module      40
$\Sigma-Mod$ category of $\Sigma$-modules      40
$\Sigma^{+}$-module      249
$\Sigma^{+}-Mod$ category of cyclic $\Sigma$-modules      249
$\Sigma^{+}_{n}$ permutation group of {0,...,n}      264
$\Sigma_{n}$ permutation group of {1,...,n}      40
$\sqcap$ box product on $Col_{K}$      86
$\sqcap$ box product on $\Sigma$-Mod      68
$\sqcap$-degree      149
$\sqcap$-differential      149
$\sqcap$-product      68
$\sqcap_{K}$-product      86
$\tau_{n}$ cycle $(0,...,n)\in\Sigma^{+}_{n}$      247
$\Theta$-algebra      7
$\Theta$-space      7
$\underline{Mon}$ non-$\Sigma$ operad for associative topological monoids      112
$\underline{W}(\underline{\mathcal{P}})$ non-$\Sigma$ W-construction on a non-$\Sigma$ operad $\underline{\mathcal{P}}$      111
$\underline{\mathcal{A}ss}$ non-$\Sigma$ operad for associative algebras      14 45
$\underline{\mathcal{A}ss}_{\infty}$ non-$\Sigma$ operad for $A_{\infty}$-algebras      13 26 106
$\underline{\mathcal{N}}$ operad $\{\underline{\mathcal{N}}(n)\}_{n\ge1}$      23 25
$\underline{\mathcal{N}}(n)$ compactification of the space $\mathcal{N}(n)$      25
$\underline{\mathcal{T}ree}$ non-$\Sigma$ operad of planar rooted trees      8 54
$\underline{\mathcal{T}ree}(n)$ set of isomorphism classes of planar rooted n-trees      8
$\uparrow U$ suspension of a graded vector space U      314
$\widehat{\mathcal{M}}$ unstable modular operad ${\{\widehat{\mathcal{M}}_{0}(g,n)\}_{g\geq0,n\geq-1}$      267
$\widehat{\mathcal{M}}(g,n)$ moduli space of genus g Riemann surfaces with n+1 labeled parametrized holes      267 304
$\widehat{\mathcal{M}}_{0}$ operad ${\{\widehat{\mathcal{M}}_{0}(n)\}_{n\geq1}$      23 207 247
$\widehat{\mathcal{{M}}_{0}(n)$ moduli space of Riemann spheres with n+1 labeled parametrized holes      23 207 267
$|\Gamma|$ geometric realization of a graph $\Gamma$      269
(n+1)-star      250
1 trivial operad      42 187
1-brace      183
2-brace      183
a(v) number of inputs (arity) of a vertex v of a rooted tree      83 217
Action of PROP      5
Additive algebraic K-theory      260
Aff group of affine transformations of the complex plane $\mathbb{C}$      213
Aff(V) group of affine transformations of a vector space V      220
Algebra over a PROP      5
Algebra over a triple      88
Algebra over operad      7
Algebraic stack      311
Algebraic theory      7
Analyseur      4
Analyseur incomplet      4
Antibracket      25
Anticyclic cooperad      262
Anticyclic operad      258
Antifield      25
Antighost      25
Antighost operator      24
Approximation      94
Arity      50 268
Associahedron      9
Associativity constraint      37
Associator      37
Atlas      311
Augmented operad      187
Balanced $A_{\infty}$-algebra      19 172 196
Bar construction on a $\mathcal{P}$-algebra      173
Bar construction on an operad      20
Batalin — Vilkovisky algebra      206
Bicolored operad      85 115
Bilinear space      254
Block permutation      40
Blowing down map      240
Brace      183
Braided monoidal category      38
Browder operation      103
Brownian bridge      18
BRST cohomology      24
BRST complex      23 24
BV-algebra      206
BY classifying space of an $A_{\infty}$-space Y      11
c(n) n-corolla      251
Canonical element      319
Canonical resolution      179
Categorical cobar complex      150
Category with multiplication      37
Center of a little disk      205
CFT      23
Chevalley — Eilenberg homology      260
Closed string field theory      25
Coarse moduli space      311
Cobar bicomplex of an operad      124
Cobar complex of an operad      20 125
Coboundary      283
Cocycle      282
Coderivation of a $\mathcal{P}$-coalgebra      169
Coefficient system      281
Coendomorphism operad      43
Cofibrant model      28
Cofree nilpotent $\mathcal{P}$-coalgebra      166
Coherence      37
Cohomological field theory      306
Col category of collections      85
Collection      85
Color of a vertex      72
Colored operad      85
Commutative $A_{\infty}$-algebra      172 196
Comp algebra      6
Complete algebraic system      3
Composition along a graph      274
Composition product      105
Con(M,n) configuration space of n distinct labeled points in a manifold M      234
Con(V,n) configuration space of n distinct labeled points in a vector space V      220
Configuration pseudo-operad      216
Conformal field theory      22 24
Connes — Tsygan exact sequence      266
Corolla      10 50
CSFT      23
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