Operads and PROPs

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

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Wheeled PROPs, graph complexes and the master equation (2006, background): “The difference between biased and unbiased definitions is explained in [13], see also a remark in Section 3 of [15].”
Generalized Operads and Their Inner Cohomomorphisms (2006, methodology): “where in particular operads with values in various categories of algebras are considered. This is partly motivated by the quantum cohomology operad. Finally, in Appendix we brieﬂy treat Markl’s list ([Mar], p. 45) in terms of labeled graphs and functors on them. Acknowledgements. Yu. M. gratefully a”
The L ∞ -deformation complex of diagrams of algebras (2009, background): “8 (= C-colored version of Proposition 57 in [35]).”
Operadic Twisting -- with an application to Deligne's conjecture (2012, background): “For more details about this subtlety, see paper [23] in which a pseudo-operad is called a non-unital Markl’s operad.”
A Foundation for Props, Algebras, and Modules (2015, background): “As discussed by Markl in [Mar08], operads, PROPs, and wheeled PROPs can all be described using collections of graphs he called pasting schemes, in these cases consisting of the level trees, the directed cycle-free graphs, and the directed graphs, respectively.”