Andrew Tonks

Homotopy Batalin-Vilkovisky Algebras

This paper provides an explicit cobrant resolution of the operad encoding Batalin{Vilkovisky algebras. Thus it denes the notion of homotopy Batalin{Vilkovisky algebras with the required homotopy properties. To dene this resolution, we extend the theory of Koszul duality to operads and properads that are dened by quadratic and linear relations. The operad encoding Batalin{Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincar e{Birkho{Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal eld theory carries a homotopy BV- algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cobrant resolution of the operad BV . We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian{Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

The 1-type of a Waldhausen K-theory spectrum

Abstract We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.

Decomposition spaces, incidence algebras and Möbius inversion I: basic theory

Abstract This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and Mobius inversion, where algebraic identities are realised by taking homotopy cardinality of equivalences of ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in . Just as the Segal condition expresses composition, the new exactness condition expresses decomposition, and there is an abundance of examples in combinatorics. After establishing some basic properties of decomposition spaces, the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra, spanned by the space of 1-simplices, and with coefficients in ∞-groupoids. We take a functorial viewpoint throughout, emphasising conservative ULF functors; these induce coalgebra homomorphisms. Reduction procedures in the classical theory of incidence coalgebras are examples of this notion, and many are examples of decalage of decomposition spaces. An interesting class of examples of decomposition spaces beyond Segal spaces is provided by Hall algebras: the Waldhausen S • -construction of an abelian (or stable infinity) category is shown to be a decomposition space. In the second paper in this series we impose further conditions on decomposition spaces, to obtain a general Mobius inversion principle, and to ensure that the various constructions and results admit a homotopy cardinality. In the third paper we show that the Lawvere–Menni Hopf algebra of Mobius intervals is the homotopy cardinality of a certain universal decomposition space. Two further sequel papers deal with numerous examples from combinatorics. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov [17] who call them unital 2-Segal spaces. Our theory is quite orthogonal to theirs: the definitions are different in spirit and appearance, and the theories differ in terms of motivation, examples, and directions.

Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees

We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids.

Spaces of maps into classifying spaces for equivariant crossed complexes

Abstract We give an equivariant version of the homotopy theory of crossed complexes. The applications generalize work on equivariant Eilenberg—Mac Lane spaces, including the non abelian case of dimension 1, and on local systems. It also generalizes the theory of equivariant 2-types, due to Moerdijk and Svensson. Further, we give results not just on the homotopy classification of maps but also on the homotopy types of certain equivariant function spaces.

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

Abstract This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Mobius inversion, with coefficients in ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition. In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control simplicial nondegeneracy. For complete decomposition spaces we establish a general Mobius inversion principle, expressed as an explicit equivalence of ∞-groupoids. Next we analyse two finiteness conditions on decomposition spaces. The first, that of locally finite length, guarantees the existence of the important length filtration for the associated incidence coalgebra. We show that a decomposition space of locally finite length is actually the left Kan extension of a semi-simplicial space. The second finiteness condition, local finiteness, ensures we can take homotopy cardinality to pass from the level of ∞-groupoids to the level of Q -vector spaces. These three conditions — completeness, locally finite length, and local finiteness — together define our notion of Mobius decomposition space, which extends Lerouxs notion of Mobius category (in turn a common generalisation of the locally finite posets of Rota et al. and of the finite decomposition monoids of Cartier–Foata), but which also covers many coalgebra constructions which do not arise from Mobius categories, such as the Faa di Bruno and Connes–Kreimer bialgebras. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov [6] who call them unital 2-Segal spaces.

Homotopy Gerstenhaber Structures and Vertex Algebras

We provide a simple construction of a G ∞ -algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of G ∞  algebras on a Calabi–Yau manifold M, extending the operations of multiplication and bracket of functions and vector fields on M, and of a Lie ∞  structure related to the bracket of Courant (Trans Amer Math Soc 319:631–661, 1990).

On the Eilenberg–Zilber theorem for crossed complexes

Abstract We give a natural strong deformation retraction from the fundamental homotopy crossed complex of a product of simplicial sets onto the tensor product of the corresponding crossed complexes. This generalises the classical theorem of Eilenberg and Zilber to a non-abelian setting. Explicit crossed complex homomorphisms analogous to the shuffle and Alexander–Whitney chain maps are given. We proceed to give a simplicially–enriched structure to the category of crossed complexes and to the simplical nerve N:Crs→S, but note that the fundamental crossed complex functor adjoint to N has only a lax or simplicially coherent enrichment.

Cubical groups which are kan

Abstract The category of cubical sets with connections of Brown and Higgins is introduced as a possible alternative to simplicial complexes for carrying out a programme of combinatorial homotopy theory. This paper proves a result crucial to the development of this theory, that group objects in the category of cubical sets with connections have the property of being Kan.

Spaces of maps into classifying spaces for equivariant crossed complexes, II: The general topological group case.

The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg-Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces.