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Joachim Kock

Publications and source records attributed to Joachim Kock.

At least 19 recordsLinked to original sources
arXiv2026

Operadic categories as (pseudo)-simplicial groupoids

From any operadic category O we construct a simplicial groupoid X (slightly pseudo in a specific way), called the operadic nerve. It integrates all the structure of chosen-local-terminals, fibre functor, and cardinality functor into a single simplicial groupoid, which can be seen as an undecking of the ordinary nerve of O in the Kleisli category for the symmetric-monoidal-groupoid monad S: we have the equation DX = SNO, where D is upper decalage. The construction leads to a new characterisation of operadic categories, in which all the axioms end up as simplicial identities, and where the notion of operad over an operadic category takes the form of a simplicial map subject to well-known pullback conditions (the notion of IKEO map).

math.CT↗
arXiv2026

Groupoid G-spans and matrices over group rings

When G is a finite abelian group, we define G-spans of groupoids and their associated matrices with entries in the group ring QG and show that composition of spans corresponds to multiplication of matrices.

math.CT↗
arXiv2025

Pita factorisation in operadic categories

In strictly factorisable operadic categories, every morphism $f$ factors uniquely as $f=\eta_f \circ \pi_f$ where $\eta_f$ is order-preserving and $\pi_f$ is a quasibijection that is order-preserving on the fibres of $\eta_f$. We call it the pita factorisation. In this paper we develop some general theory to compensate for the fact that generally pita factorisations do not form an orthogonal factorisation system. The main technical result states that a certain simplicial object in Cat, called the pita nerve, is oplax (rather than strict as it would be for an orthogonal factorisation system). The main application is the result that the so-called operadic nerve of any operadic category is coherent. This result is a key ingredient in the simplicial approach to operadic categories developed in the `main paper' [arXiv:2606.15671], which motivated the present paper. We also show that in the important case where quasibijections are invertible, the pita nerve is a decomposition space (a.k.a.~$2$-Segal space).

math.CT↗
arXiv2025

Free loop spaces and the Cauchy--Frobenius Lemma

We upgrade the Cauchy--Frobenius Lemma (`Burnside's Lemma') to a homotopy equivalence of $\infty$-groupoids, essentially given by double counting/Fubini in the free loop space of the quotient.

math.AT↗
arXiv2025

Abacus bicomodule configurations and the Bergner-Osorno-Ozornova-Rovelli-Scheimbauer equivalence

A theorem of Bergner, Osorno, Ozornova, Rovelli, and Scheimbauer states an equivalence between 2-Segal spaces and certain augmented stable double Segal spaces. In this paper we establish more general equivalences, involving simplicial maps of 2-Segal spaces and abacus bicomodule configurations, extending results of Carlier. The BOORS equivalence is recovered from the special case of the identity map. One main ingredient is an analysis of the relationship between the BOORS and Carlier notions of augmentation, hitherto considered unrelated.

math.CT↗
arXiv2024

Decomposition spaces in Combinatorics

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.

math.CO↗
arXiv2024

Convex decomposition spaces and Crapo complementation formula

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.

math.CT↗
arXiv2024

Noncrossing arithmetic

Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in free probability, it also possesses an elegant, purely combinatorial core that is of independent interest. The present article aims at offering a simple account of various aspects of higher-order Kreweras complementation on the basis of elementary arithmetic, (co)algebraic, categorical and simplicial properties of noncrossing partitions. The main idea is to consider noncrossing partitions as providing an interesting noncommutative analogue of the interplay between the divisibility poset and the multiplicative monoid of positive integers. Just as the divisibility poset can be regarded as the decalage of the multiplicative monoid, we exhibit the lattice of noncrossing partitions as the decalage of a partial monoid structure on noncrossing partitions encoding higher-order Kreweras complements. While our results may be considered familiar, several of the viewpoints can be regarded as novel, offering an efficient approach both conceptually and computationally.

math.CO↗
arXiv2023

Whole-grain Petri nets and processes

We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in the style of Meseguer and Montanari, in terms of free coloured props, and allows the following unification: for P a Petri net, the Segal space of P-processes is shown to be the free coloured prop-in-groupoids on P. There is also an unfolding semantics à la Winskel, which bypasses the classical symmetry problems: with the new formalism, every Petri net admits a universal unfolding, which in turn has associated an event structure and a Scott domain. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native. (Collective-token semantics emerges from rather drastic quotient constructions à la Best-Devillers, involving taking π_0 of the groupoids of states.)

cs.LO↗
arXiv2022

Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)

Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommutative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic rewriting systems such as chemical reaction networks.

cs.LO↗
arXiv2022

Free decomposition spaces

We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by their inert maps. We show that left Kan extension along the inclusion $j \colon \Delta_{\operatorname{inert}} \to \Delta$ takes general objects to M\"obius decomposition spaces and general maps to CULF maps. We establish an equivalence of $\infty$-categories $\mathbf{PrSh}(\Delta_{\operatorname{inert}}) \simeq \mathbf{Decomp}_{/B\mathbb{N}}$. Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions.

math.CT↗
arXiv2022

Culf maps and edgewise subdivision

We show that, for any simplicial space $X$, the $\infty$-category of culf maps over $X$ is equivalent to the $\infty$-category of right fibrations over $\operatorname{sd}(X)$, the edgewise subdivision of $X$. (When $X$ is a Rezk complete Segal or 2-Segal space, $\operatorname{sd}(X)$ is the twisted arrow category of $X$.) We give two proofs of independent interest; one exploiting comprehensive factorization and the natural transformation from the edgewise subdivision to the nerve of the category of elements, and another exploiting a new factorization system of ambifinal and culf maps, together with the right adjoint to edgewise subdivision. Using this main theorem, we show that the $\infty$-category of decomposition spaces and culf maps is locally an $\infty$-topos.

math.AT↗
arXiv2022

$\infty$-operads as symmetric monoidal $\infty$-categories

We use Lurie's symmetric monoidal envelope functor to give two new descriptions of $\infty$-operads: as certain symmetric monoidal $\infty$-categories whose underlying symmetric monoidal $\infty$-groupoids are free, and as certain symmetric monoidal $\infty$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $\infty$-operads, as a localization of a presheaf $\infty$-category, and we use this to give a simple proof of the equivalence between Lurie's and Barwick's models for $\infty$-operads.

math.CT↗
arXiv2021

The incidence comodule bialgebra of the Baez-Dolan construction

Starting from any operad P, one can consider on one hand the free operad on P, and on the other hand the Baez--Dolan construction on P. These two new operads have the same space of operations, but with very different notions of arity and substitution. The main result of this paper is that the incidence bialgebras of the two-sided bar constructions of the two operads constitute together a comodule bialgebra. The result is objective: it concerns comodule-bialgebra structures on groupoid slices, and the proof is given in terms of equivalences of groupoids and homotopy pullbacks. Comodule bialgebras in the usual sense are obtained by taking homotopy cardinality. The simplest instances of the construction cover several comodule bialgebras of current interest in analysis. If P is the identity monad, then the result is the Faà di Bruno comodule bialgebra (dual to multiplication and substitution of power series). If P is any monoid $Ω$ (considered as a one-coloured operad with only unary operations), the resulting comodule bialgebra is the dual of the near-semiring of $Ω$-moulds under product and composition, as employed in Écalle's theory of resurgent functions in local dynamical systems. If P is the terminal operad, then the result is essentially the Calaque--Ebrahimi-Fard--Manchon comodule bialgebra of rooted trees, dual to composition and substitution of B-series in numerical analysis (Chartier--Hairer--Vilmart). The full generality is of interest in category theory. As it holds for any operad, the result is actually about the Baez--Dolan construction itself, providing it with a new algebraic perspective.

math.QA↗
arXiv2020

$\infty$-Operads as Analytic Monads

We develop an $\infty$-categorical version of the classical theory of polynomial and analytic functors, initial algebras, and free monads. Using this machinery, we provide a new model for $\infty$-operads, namely $\infty$-operads as analytic monads. We justify this definition by proving that the $\infty$-category of analytic monads is equivalent to that of dendroidal Segal spaces, known to be equivalent to the other existing models for $\infty$-operads.

math.AT↗
arXiv2020

Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations

We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of Möbius inversion on the lattice of noncrossing partitions, and on the other side the more recent non-commutative shuffle-algebra approach, where the moment-cumulant relations take the form of certain exponential-logarithm relations. We achieve this by exhibiting two operad structures on (noncrossing) partitions, different in nature: one is an ordinary, non-symmetric operad whose composition law is given by insertion into gaps between elements, the other is a coloured, symmetric operad with composition law expressing refinement of blocks. We show that these operad structures interact so as to make the corresponding incidence bialgebra of the former a comodule bialgebra for the latter. Furthermore, this interaction is compatible with the shuffle structure and thus unveils how the two approaches are intertwined. Moreover, the constructions and results are general enough to extend to ordinary set partitions.

math.CO↗
arXiv2020

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

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.

math.CT↗