Koszul duality for PROPs

By Bruno Vallette. 2004

193 citations recorded by the source corpus.

The notion of PROP models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. We prove a Koszul duality theory for PROPs generalizing the one for associative algebras and for operads.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/32b62c999cef97401598a9a88a55b48ab8272f9f

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In Algebra and Number Theory

1. Jean-Louis Loday
2. Yael Fregier
3. Donald Yau
4. Joan Millès
5. Joseph Hirsh
6. Benoit Fresse
7. Martin Markl
8. Sergey Shadrin

In Geometry and Topology

1. Jean-Louis Loday
2. Donald Yau
3. Yael Fregier
4. Sergey Shadrin
5. Joan Millès
6. Joseph Hirsh
7. Martin Markl
8. Benoit Fresse

In Mathematical Physics

1. Jean-Louis Loday
2. Donald Yau
3. Yael Fregier
4. Joan Millès
5. Joseph Hirsh
6. Sergey Shadrin
7. Benoit Fresse
8. Martin Markl

Citations in the corpus

• The L ∞ -deformation complex of diagrams of algebras (2009, methodology): “When P is a properad [50], there is another approach to studying L∞deformations of P-algebras due to Merkulov and Vallette [39].”
• Homotopy theory for algebras over polynomial monads (2013, unknown): “We refer the reader to Markl [46] and JohnsonYau [32] for an explicit link with the original definitions of properads by Vallette [63] and of PROP’s by Adams-Mac Lane.”
• Curved Koszul duality theory (2010, methodology): “We summarize the different generalizations of the Koszul duality theory in the following table: ` ` ` ` ` ` ` ` ` ` ` ` Monoids Relations Homogeneous quadratic Quadratic and linear Quadratic, linear and constant Associative algebras [Pri70] [Pos93, PP05] Operads [GJ94, GK94] [GCTV09] Section 4 of th”
• Props in model categories and homotopy invariance of structures (2008, background): “Since then, several variants of the notion of operad and prop (cyclic operads [14], modular operads [15], properads [43, 44], wheeled props [35]) have been introduced.”
• Homology of generalized partition posets (2004, methodology): “i-isomorphism for any l and any m. Proof. The proof of B. Fresse lies on homological comparison lemmas. We have generalized them in the weight graded case (and for a more general monoidal product) in [V] at chapiter 5. Using the same methods than B. Fresse at 4.6 with the weight graded versions of t”
• Generalized bialgebras and triples of operads (2006, methodology): “th the symmetric group actions and to satisfy some obvious associativity and unitality axioms. The whole structure is called a prop (also 18 CHAPTER 1. ALGEBRAIC OPERADS denoted previously PROP), cf. [57, 19, 79]. An algebra (or better a gebra) over a prop is a vector space H equipped with maps P(m,”
• Wheeled PROPs, graph complexes and the master equation (2006, methodology): “The following proposition follows from [15, Section 8], see also [27,28].”