Simplicial Objects Research Articles

Principal bundle groupoids, their gauge group and their nerve

We consider groupoids in the category of principal bundles, which we call principal bundles (PB) groupoids. Inspired by work by Th. Nikolaus and K. Waldorf, we generalise bundle gerbes over manifolds to bundle gerbes over groupoids and discuss a functorial correspondence between PB groupoids and bundle gerbes over groupoids. From a PB groupoid over a fibre product groupoid, we build a bundle gerbe over another fibre product groupoid. Conversely, from a bundle gerbe over a Lie groupoid, we build a PB groupoid. It has a trivial base and from any PB groupoid with trivial base, we build a bundle gerbe over a Lie groupoid. In that case, the resulting bundle gerbe is isomorphic as a groupoid to a partial quotient of the PB groupoid. We describe the nerves of PB groupoids and their partial quotients, which are simplicial objects in the category of principal bundles. Applying this construction enables us to define the inner transformation group of the nerve of a partial quotient groupoid and to describe the transformations of the corresponding bundle gerbe.

Universal Causality

Universal Causality is a mathematical framework based on higher-order category theory, which generalizes previous approaches based on directed graphs and regular categories. We present a hierarchical framework called UCLA (Universal Causality Layered Architecture), where at the top-most level, causal interventions are modeled as a higher-order category over simplicial sets and objects. Simplicial sets are contravariant functors from the category of ordinal numbers Δ into sets, and whose morphisms are order-preserving injections and surjections over finite ordered sets. Non-random interventions on causal structures are modeled as face operators that map n-simplices into lower-level simplices. At the second layer, causal models are defined as a category, for example defining the schema of a relational causal model or a symmetric monoidal category representation of DAG models. The third layer corresponds to the data layer in causal inference, where each causal object is mapped functorially into a set of instances using the category of sets and functions between sets. The fourth homotopy layer defines ways of abstractly characterizing causal models in terms of homotopy colimits, defined in terms of the nerve of a category, a functor that converts a causal (category) model into a simplicial object. Each functor between layers is characterized by a universal arrow, which define universal elements and representations through the Yoneda Lemma, and induces a Grothendieck category of elements that enables combining formal causal models with data instances, and is related to the notion of ground graphs in relational causal models. Causal inference between layers is defined as a lifting problem, a commutative diagram whose objects are categories, and whose morphisms are functors that are characterized as different types of fibrations. We illustrate UCLA using a variety of representations, including causal relational models, symmetric monoidal categorical variants of DAG models, and non-graphical representations, such as integer-valued multisets and separoids, and measure-theoretic and topological models.

Spectral sequences in (∞,1)-categories

We explain how to set up the homotopy spectral sequence of a (co)simplicial object in an $\infty$-category, with an emphasis on how to construct the differentials in a model-invariant manner.

Simplicial polygroups and the generalized Moore complexes

A simplicial group is a simplicial object in the category of groups. A very nice application of simplicial group which is simplicial polygroup is given in this paper. Using polygroups instead of groups, we already had very good results from the well known properties due to Loday. Loday proved that a crossed module, a cat1-group, a group object in the category of categories and a simplicial group whose Moore complex is of length one are equivalent. Using Loday’s idea we present a functor from the category of groups to the category of polygroups and the simplicial groups to the simplicial polygroups. We show that there exist a functor from the category of cat1-polygroups to the category of groups and the category of groups to the category of polygroups. We also prove that the category of simplicial groups is equivalent to the category of simplicial polygroups and the category of simplicial polygroups with generalized Moore complex with of length one is equivalent to the category of polygroups.

Homotopy coherent theorems of Dold–Kan type

We establish a large class of homotopy coherent Morita-equivalences of Dold–Kan type relating diagrams with values in any weakly idempotent complete additive ∞-category; the guiding example is an ∞-categorical Dold–Kan correspondence between the ∞-categories of simplicial objects and connective coherent chain complexes.Our results generalize many known 1-categorical equivalences such as the classical Dold–Kan correspondence, Pirashvili's Dold–Kan type theorem for abelian Γ-groups and, more generally, the combinatorial categorical equivalences of Lack and Street.

The effective model structure and -groupoid objects

AbstractFor a category$\mathcal {E}$with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in$\mathcal {E}$, generalising the Kan–Quillen model structure on simplicial sets. We then prove that the effective model structure is left and right proper and satisfies descent in the sense of Rezk. As a consequence, we obtain that the associated$\infty $-category has finite limits, colimits satisfying descent, and is locally Cartesian closed when$\mathcal {E}$is but is not a higher topos in general. We also characterise the$\infty $-category presented by the effective model structure, showing that it is the full sub-category of presheaves on$\mathcal {E}$spanned by Kan complexes in$\mathcal {E}$, a result that suggests a close analogy with the theory of exact completions.

2-Segal objects and algebras in spans

We define a category parameterizing Calabi–Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in Span(C); and secondly: 2-Segal cyclic objects in C to Calabi–Yau algebra objects in Span(C).

Combinatorial parametrised spectra

We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric stabilisation machines. By means of a Grothendieck construction for model categories, we produce combinatorial model categories controlling the totality of parametrised stable homotopy theory. The global model category of parametrised symmetric spectra is equipped with a symmetric monoidal model structure (the external smash product) inducing pairings in twisted cohomology groups. As an application of our results we prove a tangent prolongation of Simpson's theorem, characterising tangent $\infty$-categories of presentable $\infty$-categories as accessible localisations of $\infty$-categories of presheaves of parametrised spectra. Applying these results to the homotopy theory of smooth $\infty$-stacks produces well-behaved (symmetric monoidal) model categories of smooth parametrised spectra. These models provide a concrete foundation for studying twisted differential cohomology, incorporating previous work of Bunke and Nikolaus.

Integration of the Kenzo System within SageMath for New Algebraic Topology Computations

This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before.

Fundamental groupoids for simplicial objects in Mal'tsev categories

We show that the category of internal groupoids in an exact Mal'tsev category is reflective, and in fact a Birkhoff subcategory of the category of simplicial objects. We then characterize the central extensions of the corresponding Galois structure, and show that regular epimorphisms admit a relative monotone-light factorization system in the sense of Chikhladze. We also draw some comparison with Kan complexes. By comparing the reflections of simplicial objects and reflexive graphs into groupoids, we exhibit a connection with weighted commutators (as defined by Gran, Janelidze and Ursini).

Simplisel Leibniz Cebirler Üzerine

Bu makalenin temel amacı, Leibniz cebirler kategorisinde simplisel objeyi tanımlayarak, simplisel Leibniz cebirler kategorisi ile Leibniz cebirler üzerinde çaprazlanmış modüller kategorisinin denkliğini göstermektir.

The unit of the total décalage adjunction

We consider the décalage construction $${{\,\mathrm{Dec}\,}}$$ and its right adjoint $$T$$. These functors are induced on the category of simplicial objects valued in any bicomplete category $${\mathcal {C}}$$ by the ordinal sum. We identify $$T{{\,\mathrm{Dec}\,}}X$$ with the path object $$X^{\Delta [1]}$$ for any simplicial object X. We then use this formula to produce an explicit retracting homotopy for the unit $$X\rightarrow T{{\,\mathrm{Dec}\,}}X$$ of the adjunction $$({{\,\mathrm{Dec}\,}},T)$$. When $${\mathcal {C}}$$ is a category of objects of an algebraic nature, we then show that the unit is a weak equivalence of simplicial objects in $${\mathcal {C}}$$.

Contractible simplicial objects

We raise the question of when a simplicial object in a catetgory is deemed contractible. The literature offers three definitions. One is the existence of an ``extra degeneracy'', indexed by $-1$, which does not quite live up to the name. This can be strengthened to a ``strong extra degeneracy". Another possibility is that it be homotopic to a constant simplicial object. Despite claims in the literature to the contrary, we show that all three are distinct concepts with strong extra degeneracy implies extra degeneracy implies homotopic to a constant and give explicit examples to show the converses fail.

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

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.

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial 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 first part of two papers, we concentrate on the simplicial and operadic aspects.

A Kenzo interface for algebraic topology computations in SageMath

In this work we present an interface between SageMath and Kenzo, together with an optional Kenzo package. Our work makes it possible to communicate both computer algebra programs and enhances the SageMath system with new capabilities in algebraic topology, dealing in particular with simplicial objects of infinite nature.

A simplicial complex splitting associativity

We introduce a simplicial object ({Dyckm}m≥0,Fi,Sj) in the category of non-symmetric algebraic operads, satisfying that Dyck0 is the operad of associative algebras and Dyck1 is J.-L. Loday’s operad of dendriform algebras. The dimensions of the operad Dyckm are given by the Fuss-Catalan numbers.Given a family of partially ordered sets P={Pn}n≥1 we show that, under certain conditions, the vector space spanned by the set of m-simpleces of P is a Dyckm algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of Dyckm algebras.

Monads of oplax actions are skew monoidales

Szlachányi showed that bialgebroids can be characterised using skew monoidal categories. The characterisation reduces the amount of data, structure, and properties required to define them. Lack and Street provide a bicategorical account of that same fact; they characterise quantum categories in terms of skew monoidal structures internal to a monoidal bicategory. A quantum category is an opmonoidal monad on an enveloping monoidale R∘⊗R in a monoidal bicategory. In a previous paper, we characterised opmonoidal arrows on enveloping monoidales as a simpler structure called oplax action.This is the second paper based on the author's PhD thesis. Here, motivated by the fact that opmonoidal monads are monads in the bicategory of monoidales, opmonoidal arrows, and opmonoidal cells; we prove that right skew monoidales are “monads of oplax actions”. To do so, we arrange oplax actions as the 1-simplices of a simplicial object in Cat. In nice cases this simplicial object is ought to be thought as a bicategory whose arrows are oplax actions, that is to say, it is weakly equivalent to a nerve of a bicategory. We define monads of oplax actions as simplicial maps out of the Catalan simplicial set and prove that these are in bijective correspondence with right skew monoidales whose unit has a right adjoint, no assumptions required on the ambient monoidal bicategory.

From simplicial homotopy to crossed module homotopy in modified categories of interest

Abstract We address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.

Bar simplicial modules and secondary cyclic (co)homology

In this paper we study the simplicial structure of the complex C^{\bullet}((A,B,\varepsilon); M) , associated to the secondary Hochschild cohomology. The main ingredient is the simplicial object \mathcal{B}(A,B,\varepsilon) , which plays a role equivalent to that of the bar resolution associated to an algebra. We also introduce the secondary cyclic (co)homology and establish some of its properties (Theorems 4.11 and 5.11).