Operads and PROPs

Abstract We review definitions and basic properties of operads, PROPs and algebras over these structures.

[1]  S. Maclane,et al.  Categorical Algebra , 2007 .

[2]  R. Umble.  Matrons, A∞-bialgebras and the Polytopes Kk , 2005 .

[3]  Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin-Vilkovisky algebras , 1993, hep-th/9306021.

[4]  J. May.  CATERADS AND ALGEBRAS OVER CATERADS , 2005 .

[5]  M. Markl,et al.  Algebras with one operation including Poisson and other Lie-admissible algebras , 2004, math/0412206.

[6]  A. Voronov,et al.  Notes on string topology , 2005, math/0503625.

[7]  Stephen Halperin,et al.  Lectures on minimal models , 1983 .

[8]  Ieke Moerdijk,et al.  Axiomatic homotopy theory for operads , 2002, math/0206094.

[9]  J.-L. Loday,et al.  Algèbres ayant deux opérations associatives (digèbres) , 1995 .

[10]  Michael Batanin,et al.  Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories☆ , 1998 .

[11]  Albert Burroni.  $T$-catégories (catégories dans un triple) , 1971 .

[12]  Operads, Algebras and Modules in General Model Categories , 2001, math/0101102.

[13]  R. Vogt.  Cofibrant operads and universal E∞ operads , 2003 .

[14]  M. Kontsevich,et al.  Gromov-Witten classes, quantum cohomology, and enumerative geometry , 1994 .

[15]  I. Moerdijk,et al.  RESOLUTION OF COLOURED OPERADS AND RECTIFICATION OF HOMOTOPY ALGEBRAS , 2005, math/0512576.

[16]  S. Merkulov.  PROP profile of deformation quantization , 2004 .

[17]  Alain Connes,et al.  Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem , 2000 .

[18]  J. P. May,et al.  The geometry of iterated loop spaces , 1972 .

[19]  Tom Leinster.  Higher Operads, Higher Categories , 2003 .

[20]  M. Markl.  Cyclic operads and homology of graph complexes , 1998, math/9801095.

[21]  J. M. Boardman,et al.  Homotopy Invariant Algebraic Structures on Topological Spaces , 1973 .

[22]  Raymond O. Wells,et al.  Differential analysis on complex manifolds , 1980 .

[23]  J.-L. Loday.  La renaissance des opérades , 1995 .

[24]  Operads and Moduli Spaces of Genus 0 Riemann Surfaces , 1994, alg-geom/9411004.

[25]  M. Kontsevich.  Deformation Quantization of Poisson Manifolds , 1997, q-alg/9709040.

[26]  M. Markl.  INTRINSIC BRACKETS AND THE L∞-DEFORMATION THEORY OF BIALGEBRAS , 2004 .

[27]  P. Deligne.  Résumé des premiers exposés de A. Grothendieck , 1972 .

[28]  Murray Gerstenhaber,et al.  The Cohomology Structure of an Associative Ring , 1963 .

[29]  Giovanni Gallavotti,et al.  Twistless KAM tori , 1993, chao-dyn/9306003.

[30]  M. Kapranov,et al.  Koszul duality for Operads , 1994, 0709.1228.

[31]  V. Schechtman,et al.  Homotopy Lie algebras , 1993 .

[32]  Jim Stasheff,et al.  Homotopy associativity of $H$-spaces. II , 1963 .

[33]  Bruno Vallette,et al.  Dualité de Koszul des PROPs , 2004, math/0405057.

[34]  A. Voronov.  Notes on universal algebra , 2001, math/0111009.

[35]  N. Strickland,et al.  BRAVE NEW WORLDS IN STABLE HOMOTOPY THEORY , 1997 .

[36]  M. Batanin.  The Eckmann-Hilton argument, higher operads and En-spaces. , 2002 .

[37]  I. Kríz,et al.  Operads, algebras, modules and motives , 2018, Astérisque.

[38]  A. Voronov,et al.  On operad structures of moduli spaces and string theory , 1993, hep-th/9307114.

[39]  Michael A. Mandell.  E algebras and p-adic homotopy theory , 2001 .

[40]  S. Merkulov.  PROP Profile of Poisson Geometry , 2004, math/0401034.

[41]  Ezra Getzler,et al.  Cyclic Operads and Cyclic Homology , 1995 .

[42]  M. Kontsevich.  FORMAL (NON)-COMMUTATIVE SYMPLECTIC GEOMETRY , 1993 .

[43]  D. Sullivan,et al.  String Topology , 1999, math/9911159.

[44]  The Combinatorics of Iterated Loop Spaces , 2003, math/0301221.

[45]  S. Saneblidze,et al.  The Biderivative and a ∞ -bialgebras , 2004 .

[46]  Ezra Getzler,et al.  Homotopy algebra and iterated integrals for double loop spaces , 1994 .

[47]  I. Kríz,et al.  Higher String Topology on General Spaces , 2022 .

[48]  S. Eilenberg,et al.  Adjoint functors and triples , 1965 .

[49]  J. May.  DEFINITIONS: OPERADS, ALGEBRAS AND MODULES , 2002 .

[50]  Tom Leinster.  Operads in Higher-Dimensional Category Theory , 2000 .

[51]  Martin Markl.  Loop Homotopy Algebras in Closed String Field Theory , 2001 .

[52]  A resolution (minimal model) of the PROP for bialgebras , 2002, math/0209007.

[53]  I. Kríz,et al.  On Kontsevich's Hochschild cohomology conjecture , 2003, Compositio Mathematica.

[54]  M. Markl.  Cotangent cohomology of a category and deformations , 1996 .

[55]  Martin Markl.  A Compactification of the Real Configuration Space as an Operadic Completion , 1996 .

[56]  J. M. Boardman,et al.  Homotopy-everything $H$-spaces , 1968 .

[57]  Another proof of M. Kontsevich formality theorem , 1998, math/9803025.

[58]  Peter Hilton,et al.  A Course in Homological Algebra , 1972 .

[59]  Deformations of algebras over operads and Deligne's conjecture , 2000, math/0001151.

[60]  B. Shoikhet.  The CROCs, non-commutative deformations, and (co)associative bialgebras , 2003, math/0306143.

[61]  R. Vogt,et al.  The categories ofA∞- andE∞-monoids and ring spaces as closed simplicial and topological model categories , 1991 .

[62]  D. Mumford,et al.  The irreducibility of the space of curves of given genus , 1969 .

[63]  Vladimir Hinich.  Homological algebra of homotopy algebras , 1997 .

[64]  Teimuraz Pirashvili.  On the PROP corresponding to bialgebras , 2001 .

[65]  Saunders MacLane,et al.  Natural Associativity and Commutativity , 1963 .

[66]  Joachim Lambek.  Deductive systems and categories , 2005, Mathematical systems theory.

[67]  BRUNO VALLETTE.  A Koszul duality for props , 2007 .

[68]  P. Salvatore,et al.  Configuration spaces with summable labels , 1999, math/9907073.

[69]  Dennis Sullivan,et al.  Infinitesimal computations in topology , 1977 .

[70]  Frédéric Chapoton,et al.  Dialgebras and Related Operads , 2001 .

[71]  An explicit deformation theory of (co)associative bialgebras , 2003, math/0310320.

[72]  Homotopy Algebras are Homotopy Algebras , 1999, math/9907138.

[73]  D. Quillen,et al.  Cyclic homology and the Lie algebra homology of matrices , 1984 .

[74]  V. Hinich.  Tamarkin's proof of Kontsevich formality theorem , 2000, math/0003052.

[75]  Satyan L. Devadoss.  Tessellations of Moduli Spaces and the Mosaic Operad , 1998, math/9807010.

[76]  F. Knudsen.  The projectivity of the moduli space of stable curves, II: The stacks $M_{g,n}$ , 1983 .

[77]  Ioan Mackenzie James,et al.  Handbook of algebraic topology , 1995 .

[78]  Vladimir Hinich,et al.  Cyclic operads and algebra of chord diagrams , 2000, math/0005197.

[79]  M. Markl.  Distributive laws and Koszulness , 1996 .

[80]  Gian-Carlo Rota,et al.  Coalgebras and Bialgebras in Combinatorics , 1979 .

[81]  M. M. KAPRANOVwhere.  Modular Operads , 1994 .

[82]  Martin Markl.  Models for operads , 1994 .

[83]  On the invertibility of quantization functors , 2003, math/0306212.

[84]  P. Deligne,et al.  Groupes de monodromie en geometrie algebrique , 1972 .

[85]  M. Batanin.  Homotopy coherent category theory and A∞-structures in monoidal categories , 1998 .

[86]  Wee Liang Gan.  Koszul Duality for Dioperads , 2022 .

[87]  Yuri I. Manin,et al.  Frobenius manifolds, quantum cohomology, and moduli spaces , 1999 .

[88]  Homology and cohomology with coefficients, of an algebra over a quadratic operad , 1998 .

[89]  D. Mumford,et al.  The projectivity of the moduli space of stable curves. I: Preliminaries on "det" and "Div". , 1976 .

[90]  Michael A. Mandell.  Cochains and homotopy type , 2003, math/0311016.

[91]  Nicholas Hamblet,et al.  The Geometry of Iterated Loop Spaces , 2007 .

[92]  Yi-Zhi Huang.  Two-Dimensional Conformal Geometry and Vertex Operator Algebras , 1997 .

[93]  D. Tamarkin,et al.  Cyclic Formality and Index Theorems , 2001 .

[94]  J. May.  OPERADIC TENSOR PRODUCTS AND SMASH PRODUCTS , 1997 .

[95]  The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points , 2005, math/0507514.

[96]  Martin Markl,et al.  Operads in algebra, topology, and physics , 2002 .