arXiv subjects
Publications and source records attributed to Imma Gálvez-Carrillo.
The theory of 2-Segal sets has connections to various important constructions such as the Waldhausen $S_\bullet$-construction in algebraic $K$-theory, Hall algebras, and (co)operads. In this paper, we construct 2-Segal sets from rooted trees and explore how these applications are illustrated by this example.
A decomposition space (also called 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses composition, the new condition expresses decomposition. It is a general framework for incidence (co)algebras. In this contribution, after establishing a formula for the section coefficients, we survey a large supply of examples, emphasising the notion's firm roots in classical combinatorics. The first batch of examples, similar to binomial posets, serves to illustrate 2 key points: (1) the incidence algebra in question is realised directly from a decomposition space, without a reduction step, and reductions are often given by CULF functors; (2) at the objective level, the convolution algebra is a monoidal structure of species. We encounter the usual Cauchy product of species, the shuffle product of L-species, the Dirichlet product of arithmetic species, the Joyal-Street external product of q-species and the Morrison `Cauchy' product of q-species. In each case a power series representation results from taking cardinality. The external product of q-species exemplifies the fact that Waldhausen's S-construction on an abelian category is a decomposition space, yielding Hall algebras. The next class of examples includes Schmitt's chromatic Hopf algebra, the Faà di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf algebra of trees and variations from operad theory. Similar structures on posets and directed graphs exemplify a general construction of decomposition spaces from directed restriction species. An appetiser on decomposition spaces of symmetric functions is included. We finish by computing the Möbius function in a few cases, and commenting on certain cancellations that occur in the process of taking cardinality, substantiating that these cancellations are not possible at the objective level.
We establish a structure theorem, analogous to the classical result of Milnor and Moore, for differential graded Hopf algebras: any differential Hopf algebra $H$ that is free as a coalgebra carries an underlying $B_\infty$ algebra structure that restricts to the subspace of primitives, and conversely $H$ may be recovered via a universal enveloping differential-2-associative algebra. This extends the work of Loday and Ronco [12] where the ungraded non-differential case was treated, and only the multibrace part of the $B_\infty$ structure was found. We show that the multibrace structure of [12] originates from a twisting of a quasi-trivial structure, extending the work of Markl [14] on the $A_\infty$ structure underlying any algebra with a square-zero endomorphism. In this framework it is also clear that the multibrace and $A_\infty$ structures are compatible, and provide an appropriate $B_\infty$ structure for the structure theorem.
We establish a Crapo complementation formula for the Möbius function $μ^X$ in a general decomposition space $X$ in terms of a convex subspace $K$ and its complement: $μ^X \simeq μ^{X\setminus K} + μ^X*ζ^K*μ^X$. We work at the objective level, meaning that the formula is an explicit homotopy equivalence of $\infty$-groupoids. Almost all arguments are formulated in terms of (homotopy) pullbacks. Under suitable finiteness conditions on $X$, one can take homotopy cardinality to obtain a formula in the incidence algebra at the level of $\mathbb{Q}$-algebras. When $X$ is the nerve of a locally finite poset, this recovers the Björner--Walker formula, which in turn specialises to the original Crapo complementation formula when the poset is a finite lattice. A substantial part of the work is to introduce and develop the notion of convexity for decomposition spaces, which in turn requires some general preparation in decomposition-space theory, notably some results on reduced covers and ikeo and semi-ikeo maps. These results may be of wider interest. Once this is set up, the objective proof of the Crapo formula is quite similar to that of Björner--Walker.
Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality.
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 Möbius inversion, with coefficients in $\infty$-groupoids. A decomposition space is a simplicial $\infty$-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 Möbius inversion principle, expressed as an explicit equivalence of $\infty$-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 on 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 $\infty$-groupoids to the level of vector spaces. These three conditions - completeness, locally finite length, and local finiteness - together define our notion of Möbius decomposition space, which extends Leroux's notion of Möbius 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 Möbius categories, such as the Faà di Bruno and Connes-Kreimer bialgebras. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who call them unital 2-Segal spaces.
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation.
This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and Möbius inversion, where algebraic identities are realised by taking homotopy cardinality of equivalences of $\infty$-groupoids. A decomposition space is a simplicial $\infty$-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in Delta. Just as the Segal condition expresses composition, the new 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 $\infty$-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. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) 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.
Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of Möbius decomposition space, a far-reaching generalisation of the notion of Möbius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of Möbius intervals, which contains the universal Möbius function (but is not induced by a Möbius category), can be realised as the homotopy cardinality of a Möbius decomposition space $U$ of all Möbius intervals, and that in a certain sense $U$ is universal for Möbius decomposition spaces and CULF functors.
By homotopy linear algebra we mean the study of linear functors between slices of the $\infty$-category of $\infty$-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into $\infty$-categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality à la Baez-Hoffnung-Walker compatible with this duality. We needed these results to support our work on incidence algebras and Möbius inversion over $\infty$-groupoids; we hope that they can also be of independent interest.
We show that Schmitt's restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce the notion of directed restriction species that subsume Schmitt's restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher-Connes-Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces.
We introduce the notion of decomposition space as a general framework for incidence algebras and Möbius inversion: it is a simplicial infinity-groupoid satisfying an exactness condition weaker than the Segal condition, which expresses decomposition. We work on the objective level of homotopy linear algebra with coefficients in infinity-groupoids, developed along the way. To any (complete) decomposition space there is associated an incidence (co)algebra (with coefficients in infinity-groupoids), shown to satisfy a sign-free version of the Möbius inversion principle. Examples of decomposition spaces beyond Segal spaces are given by the Waldhausen S-construction and by Schmitt restriction species. Imposing certain homotopy finiteness conditions yields the notion of Möbius decomposition space, an extension of the notion of Möbius category of Leroux. We take a functorial viewpoint throughout, emphasising conservative ULF functors, and show that most reduction procedures in the classical theory are examples of this notion, and in particular that many are examples of decalage of decomposition spaces. Our main theorem concerns the Lawvere-Menni Hopf algebra of Möbius intervals, which contains the universal Möbius function (but does not come from a Möbius category): we establish that Möbius intervals form a decomposition space, which is in some sense universal. NOTE: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who call it unital 2-Segal space. Our theory is quite orthogonal to theirs.
The d.g. operad C of cellular chains on the operad of spineless cacti is isomorphic to the Gerstenhaber-Voronov operad codifying the cup product and brace operations on the Hochschild cochains of an associative algebra, and to the suboperad F_2X of the surjection operad. Its homology is the Gerstenhaber operad G. We construct an operad map psi from A-infinity to C such that psi(m_2) is commutative and the homology of psi is the canonical map A \to Com \to G. This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. Our explicit A-infinty structure does not vanish on non-trivial shuffles in higher degrees, so does not give a map from Com-infinity to C. If such a map could be written down explicitly, it would immediately lead to a G-infinity structure on C and on Hochschild cochains, that is, to a direct proof of the Deligne conjecture.
We construct spectral sequences in the framework of Baues-Wirsching cohomology and homology for functors between small categories and analyze particular cases including Grothendieck fibrations. We also give applications to more classical cohomology and homology theories including Hochschild-Mitchell cohomology and those studied before by Watts, Roos, Quillen and others.