Lecture Notes in Mathematics 2147. Cham: Springer (ISBN 978-3-319-20546-5/pbk; 978-3-319-20547-2/ebook). xv, 358 p. (2015).
The theory of $$\infty$$-operads has been thoroughly developed in two equivalent models in the works of J. Lurie [“Higher algebra”, http://www.math.harvard.edu/~lurie/], on one hand and D.-C. Cisinski and I. Moerdijk [J. Topol. 4, No. 2, 257–299 (2011; Zbl 1221.55011)] on the other hand. Let us recall briefly the approach of the second, inspired by Joyal’s work on quasi-categories. Joyal defines them as simplicial sets satisfying the inner Kan condition and characterizes them from a homotopy viewpoint as the fibrant objects of an appropriate model category structure on simplicial sets. In the definition of simplicial sets as functors $$\Delta^{op}\rightarrow Set$$, Cisinski and Moerdijk replace the category $$\Delta$$ by a more general “category of trees” $$\Omega$$, generated by trees and coface and codegeneracy maps between these trees, which are the appropriate generalizations for trees of the cofaces and codegeneracies of simplices, relying on the notion of graph substitution (how to replace a vertex in a given graph by another graph). These maps amounts to collapse and add vertices in trees. Dendroidal sets are then defined as functors $$\Omega^{op}\rightarrow Set$$ forming a category $$dSet$$, and an analogue of the inner Kan condition allows one to define quasi-operads, or $$\infty$$-operads, as inner Kan dendroidal sets. It turns out that $$dSet$$ possesses a model structure whose fibrant objects are precisely the quasi-operads. Moreover, a recent work of G. Heuts, V. Hinich and I. Moerdijk [“On the equivalence between Lurie’s model and the dendroidal model for infinity-operads”, arXiv:1305.3658] establishes the equivalence between Cisinski-Moerdijk’s quasi-operads and Lurie’s $$\infty$$-operads (in the case of operads without constants).
Properads are a generalization of operads introduced by B. Vallette [Trans. Am. Math. Soc. 359, No. 10, 4865–4943 (2007; Zbl 1140.18006)] which parametrize algebraic structures with several inputs and several outputs (that is, bialgebra-like structures). Given that such structures naturally appear in various situations (Hopf algebras in representation theory, Frobenius bialgebras in algebraic topology, Lie bialgebras and their variants in mathematical physics and string topology), it is natural to ask for a generalization of the theory of $$\infty$$-operads to the properad setting. The main purpose of the present monograph is precisely to generalize in this context the approach of Cisinski-Moerdijk, by replacing the dendroidal category $$\Omega$$ by a “graphical category” $$\Gamma$$ with the appropriate generalizations of coface and codegeneracy maps, in particular the inner maps which allows to define the inner Kan condition. Let us emphasize that the construction of graphical sets and quasi-properads is not at all a straightforward generalization of the dendroidal case. Passing from trees to connected directed graphs implies a lot of technical difficulties which are nicely overcome in this book, in particular in the careful analysis of the graph substitution operations. All these constructions are the topic of Chapters 2,5, 6 and 7 of Part I. In Part II, this theory is generalized to directed connected graphs with wheels in order to define a proper notion of wheeled $$\infty$$-properads. Such structures are also interesting since adding wheels in the construction of properads is reflected, at the level of the algebraic structures it encode, by adding trace maps (see Markl-Merkulov-Shadrin [M. Markl et al., J. Pure Appl. Algebra 213, No. 4, 496–535 (2009; Zbl 1175.18002)], and S. A. Merkulov [in: European congress of mathematics. Proceedings of the 5th ECM congress, Amsterdam, Netherlands, July 14–18, 2008. Zürich: European Mathematical Society (EMS). 83–114 (2010; Zbl 1207.18010)]).
The concluding chapter 11 offers some perspectives pointing towards different directions. Let us note among these that a natural continuation of this work would be to characterize $$\infty$$-properads as fibrant objects in an appropriate model structure on graphical sets, and to compare them with the simplicial model for $$\infty$$-properads obtained by P. Hackney and M. Robertson [“The homotopy theory of simplicial props”, arXiv:1209.1087].

### MSC:

 18-02 Research exposition (monographs, survey articles) pertaining to category theory 18D50 Operads (MSC2010) 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.) 55P48 Loop space machines and operads in algebraic topology 55U10 Simplicial sets and complexes in algebraic topology 05C99 Graph theory 55U40 Topological categories, foundations of homotopy theory