arXiv subjects
Joachim Kock
Publications and source records attributed to Joachim Kock.
Decomposition-space slices are toposes
We show that the category of decomposition spaces and CULF maps is locally a topos. Precisely, the slice category over any decomposition space D is a presheaf topos, namely decomp/D=Psh(tw D).
Antipodes of monoidal decomposition spaces
We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as $μ= ζ\circ S$, just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors $S_{\textrm{even}} - S_{\textrm{odd}}$, and it is a refinement of the general Möbius inversion construction of Gálvez-Kock-Tonks, but exploiting the monoidal structure.
Operadic categories and décalage
Batanin and Markl's operadic categories are categories in which each map is endowed with a finite collection of "abstract fibres" -- also objects of the same category -- subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the décalage comonad D on small categories. A simple case involves unary operadic categories -- ones wherein each map has exactly one abstract fibre -- which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monad induced on the category of D-coalgebras by the forgetful-cofree adjunction. A similar description is found for general operadic categories arising out of a corresponding analysis that starts from a "modified décalage" comonad on the arrow category of Cat.
From Möbius inversion to renormalisation
This paper traces a straight line from classical Möbius inversion to Hopf-algebraic perturbative renormalisation. This line, which is logical but not entirely historical, consists of just a few main abstraction steps, and some intermediate steps dwelled upon for mathematical pleasure. The paper is largely expository, but contains many new perspectives on well-known results. For example, the equivalence between the Bogoliubov recursion and the Atkinson formula is exhibited as a direct generalisation of the equivalence between the Weisner--Rota recursion and the Hall--Leroux formula for Möbius inversion.
Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
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, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals
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.
Regular patterns, substitudes, Feynman categories and operads
We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras. There are three main ingredients in establishing these biequivalences. The first is a strictification theorem (exploiting Power's General Coherence Result) which allows to reduce to the case where the structure maps are identity-on-objects functors and strict monoidal. Second, we subsume the Getzler and Kaufmann--Ward hereditary axioms into the notion of Guitart exactness, a general condition ensuring compatibility between certain left Kan extensions and a given monad, in this case the free-symmetric-monoidal-category monad. Finally we set up a biadjunction between substitudes and what we call pinned symmetric monoidal categories, from which the results follow as a consequence of the fact that the hereditary map is precisely the counit of this biadjunction.
Homotopy linear algebra
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.
Decomposition spaces and restriction species
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.
Univalence in locally cartesian closed infinity-categories
After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every infinity-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (n-2)-truncated maps. We show that univalent families are preserved (and detected) by right adjoints to locally cartesian localizations, and use this to exhibit certain canonical univalent families in infinity-quasitopoi (certain infinity-categories of "separated presheaves", introduced here). We also exhibit some more exotic examples of univalent families, illustrating that a univalent family in an n-topos need not be (n-2)-truncated, as well as some univalent families in the Morel--Voevodsky infinity-category of motivic spaces, an instance of a locally cartesian closed infinity-category which is not an n-topos for any $0\leq n\leq\infty$. Lastly, we show that any presentable locally cartesian closed infinity-category is modeled by a combinatorial type-theoretic model category, and conversely that the infinity-category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed. Under this correspondence, univalent families in presentable locally cartesian closed infinity-categories correspond to univalent fibrations in combinatorial type-theoretic model categories.
Polynomial functors and combinatorial Dyson-Schwinger equations
We present a general abstract framework for combinatorial Dyson-Schwinger equations, in which combinatorial identities are lifted to explicit bijections of sets, and more generally equivalences of groupoids. Key features of combinatorial Dyson-Schwinger equations are revealed to follow from general categorical constructions and universal properties. Rather than beginning with an equation inside a given Hopf algebra and referring to given Hochschild $1$-cocycles, our starting point is an abstract fixpoint equation in groupoids, shown canonically to generate all the algebraic structure. Precisely, for any finitary polynomial endofunctor $P$ defined over groupoids, the system of combinatorial Dyson-Schwinger equations $X=1+P(X)$ has a universal solution, namely the groupoid of $P$-trees. The isoclasses of $P$-trees generate naturally a Connes-Kreimer-like bialgebra, in which the abstract Dyson-Schwinger equation can be internalised in terms of canonical $B_+$-operators. The solution to this equation is a series (the Green function) which always enjoys a Faà di Bruno formula, and hence generates a sub-bialgebra isomorphic to the Faà di Bruno bialgebra. Varying $P$ yields different bialgebras, and cartesian natural transformations between various $P$ yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to truncation of Dyson-Schwinger equations. Finally, all constructions can be pushed inside the classical Connes-Kreimer Hopf algebra of trees by the operation of taking core of $P$-trees. A byproduct of the theory is an interpretation of combinatorial Green functions as inductive data types in the sense of Martin-Löf Type Theory (expounded elsewhere).
Cospan construction of the graph category of Borisov and Manin
It is shown how the graph category of Borisov and Manin can be constructed from (a variant of) the graph category of Joyal and Kock, essentially by reversing the generic morphisms. More precisely, the morphisms in the Borisov-Manin category are exhibited as cospans of reduced covers and refinement morphisms.
Faà di Bruno for operads and internal algebras
For any coloured operad R, we prove a Faà di Bruno formula for the `connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Faà di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faa di Bruno formula for P-trees of Gálvez--Kock--Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following Gálvez--Kock--Tonks, we work at the objective level of groupoid slices, hence all proofs are `bijective': the formula is established as the homotopy cardinality of an explicit equivalence of groupoids. In fact we establish the formula more generally in a relative situation, for algebras for one polynomial monad internal to another. This covers in particular nonsymmetric operads (for which the terminal reduced case yields the noncommutative Faà di Bruno formula of Brouder--Frabetti--Krattenthaler).
Homotopy composition of cospans
It is well known that the category of finite sets and cospans, composed by pushout, contains the universal {\em special} commutative Frobenius algebra. In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.
Decomposition Spaces, Incidence Algebras and Möbius Inversion
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.
Combinatorial Dyson-Schwinger equations and inductive data types
The goal of this contribution is to explain the analogy between combinatorial Dyson-Schwinger equations and inductive data types to a readership of mathematical physicists. The connection relies on an interpretation of combinatorial Dyson-Schwinger equations as fixpoint equations for polynomial functors (established elsewhere by the author, and summarised here), combined with the now-classical fact that polynomial functors provide semantics for inductive types. The paper is expository, and comprises also a brief introduction to type theory.
Hochster duality in derived categories and point-free reconstruction of schemes
For a commutative ring $R$, we exploit localization techniques and point-free topology to give an explicit realization of both the Zariski frame of $R$ (the frame of radical ideals in $R$) and its Hochster dual frame, as lattices in the poset of localizing subcategories of the unbounded derived category $D(R)$. This yields new conceptual proofs of the classical theorems of Hopkins-Neeman and Thomason. Next we revisit and simplify Balmer's theory of spectra and supports for tensor triangulated categories from the viewpoint of frames and Hochster duality. Finally we exploit our results to show how a coherent scheme $(X,\mathcal{O}_X)$ can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes.