Simplicial Model Research Articles

Chern–Weil map for principal bundles over groupoids

The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal G-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal G-bundles. In this paper, we study differential geometry of these objects, including connections and holonomy maps. We also introduce a Chern–Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes. As an application, we recover the equivariant Chern–Weil map of Bott–Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott–Shulman map $$S(\mathfrak{g}^*)^G \to H^*(BG)$$ when the manifold is a point.

Integration of simplicial forms and Deligne cohomology

We present two approaches to constructing an integration map along the fiber for smooth Deligne cohomology. The first is defined in the simplicial model, where a class in Deligne cohomology is represented by a simplicial form, and the second in a related but more combinatorial model .

Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a ‘system of open neighbourhoods at infinity’ while an exterior map is a continuous map which is ‘continuous at infinity’. The category of spaces and proper maps is a subcategory of the category of exterior spaces.

Tower techniques for cosimplicial resolutions

Let ℳ be a simplicial model category and J : ℳ → ℳ a simplicial coaugmented functor. Given an object X, the assignment n↦J n+1 X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J s X = tots([n] ↦ J n+1 X). An object X is called J-injective if it is a retract of JX in Ho(ℳ) via the natural map. We show that certain homotopy limits of J-injective objects are J s -injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) → X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}s≥0→{totS X}s ≥ 0.

A homotopy idempotent construction by means of simplicial groups

We obtain a simplicial group model for localization of (not necessarily nilpotent) spaces at sets of primes by applying a suitable functor dimensionwise, as in earlier work of Quillen and Bousfield-Kan. For a set of primesP and any groupG, letG→L PG be a universal homomorphism fromG into a group which is uniquely divisible by primes not inP, and denote also byL P the prolongation of this functor to simplicial groups. We prove that, ifX is any connected simplicial set andJ is any free simplicial group which is a model for the loop space ΩX, then the classifying space $$\overline W LpJ$$ is homotopy equivalent to the localization ofX atP. Thus, there is a map $$X \to \overline W LpJ$$ which is universal among maps fromX into spacesY for which the semidirect products πκ(Y)⋊ π1(Y) are uniquely divisible by primes not inP. This approach also yields a neat construction of fibrewise localization.

H-spaces with cyclic symmetry

In [Topology Appl., forthcoming paper], we have proved that for a 2-antilocal H-space there exists a strictly commutative simplicial pseudogroup structure on its minimal simplicial model. Here, we show that for a p-antilocal H-space there exists a p-variable simplicial pseudogroup structure on its minimal simplicial model which has strictly cyclic symmetry.

Simplicial structures on model categories and functors

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.

Replacing model categories with simplicial ones

In this paper we show that model categories of a very broad class can be replaced up to Quillen equivalence by simplicial model categories.

On simplicial loops and H-spaces

An algebraic loop is a ‘group without associativity’. It holds that a surjective homomorphism of simplicial loops is a Kan fibration, the Moore complex is a loop-valued chain complex with homology the homotopy groups of the simplicial loop, and the simplicial loop is minimal iff the Moore complex is minimal. We show that the minimal simplicial model of a connected H-space X is a simplicial loop. As an application, we give a short proof of a Theorem of Kock, Kristensen and Madsen which represents cohomology theories by the homology of loop-valued cochain functors. Then we show that the Postnikov decomposition of a connected minimal simplicial loop consists of central loop extensions. Using this, we prove a refinement of a Theorem of Iriye and Kono: For a connected 2-antilocal H-space X, there exists a strict commutative loop structure on its minimal model.

MORE ON GENERALIZED SIMPLICIAL CHIRAL MODELS

By generalizing the auxiliary field term in the Lagrangian of simplicial chiral models on a (d-1)-dimensional simplex, the generalized simplicial chiral models has been introduced in Ref. 1. These models can be solved analytically only in d=0 and d=2 cases at large-N limit. In the d=0 case, we calculate the eigenvalue density function in strong regime and show that the partition function computed from this density function is consistent with one calculated by path integration directly. In the d=2 case, it is shown that all V= Tr (AA†)n models have a third order phase transition, the same as the two-dimensional Yang–Mills theory.

Simplicial Models for the Global Dynamics of Attractors

Given an unknown attractor A in a continuous dynamical system, how we can discover the topology and dynamics of A? As a practical matter, how can we do so from only a finite amount of information? One way of doing so is to produce a semi-conjugacy from A onto a model system M whose topology and dynamics are known. The complexity of M then provides a lower bound for the complexity of A. In this paper, we use the techniques of the Conley index to construct a simplicial model and a surjective semi-conjugacy for a large class of attractors. The essential features of this construction are that the model M can be explicitly described and that the finite amount of information needed to construct it is computable.

Presheaves of symmetric spectra

It is shown that the category of presheaves of symmetric spectra on a small Grothendieck site C admits a proper closed simplicial model structure so that the associated homotopy category is adjoint equivalent to the stable category associated to presheaves of spectra on C .

Simplicial minisuperspace models in the presence of a massive scalar field with arbitrary scalar coupling R2

We consider a simplicial minisuperspace model based on a cone over the 4 triangulation of the 3-sphere, in the presence of a massive scalar field, , with arbitrary scalar coupling term R2. By restricting all the interior edge lengths and all the boundary edge lengths to be equivalent and the scalar field to be homogeneous on the 3-space, we obtain a two-dimensional minisuperspace model {si,i} for what is one of the most relevant triangulations of the spatial universe. We solve the two classical equations and find that there are both real Euclidean and Lorentzian classical solutions for any size of the boundary 3-space, 4. After studying the analytic properties of the action in the space of complex edge lengths we then obtain steepest descents contours of constant imaginary action passing through Lorentzian classical geometries giving the dominant contribution and yielding a convergent wavefunction of the universe. We also show that the semi-classical wavefunctions for the Euclidean solutions associated with large boundary 3-spaces are exponentially suppressed.

On cohomology algebras of complex subspace arrangements

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.

Hodge decomposition for higher order Hochschild homology

Let Γ be the category of finite pointed sets and F be a functor from Γ to the category of vector spaces over a characteristic zero field. Loday proved that one has the natural decomposition π nF(S 1)≅⊕ i=0 nH n (i)(F), n≥0. We show that for any d≥1, there exists a similar decomposition for π n F( S d ). Here S d is a simplicial model of the d-dimensional sphere. The striking point is, that the knowledge of the decomposition for π n F( S 1) (respectively π n F( S 2)) completely determines the decomposition of π n F( S d ) for any odd (respectively even) d. These results can be applied to the cohomology of the mapping space X S d , where X is a d-connected space. Thus Hodge decomposition of H ∗(X S 1 ) and H ∗(X S 2 ) determines all groups H ∗(X S d ),d≥1 .

Generalized simplicial chiral models

Using the auxiliary field representation of the simplicial chiral models on a ( d−1)-dimensional simplex, the simplicial chiral models are generalized through replacing the term Tr (AA †) in the Lagrangian of these models by an arbitrary class function of AA † ; V(AA †) . This is the same method used in defining the generalized two-dimensional Yang–Mills theories (gYM 2) from ordinary YM 2. We call these models the “generalized simplicial chiral models”. Using the results of the one-link integral over a U( N) matrix, the large- N saddle-point equations for eigenvalue density function ρ( z) in the weak ( β> β c) and strong ( β< β c) regions are computed. In d=2, where the model is in some sense related to the gYM 2 theory, the saddle-point equations are solved for ρ( z) in the two regions, and the explicit value of critical point β c is calculated for V(B)= Tr B n (B=AA †) . For V(B)= Tr B 2, Tr B 3 , and Tr B 4, the critical behaviour of the model at d=2 is studied, and by calculating the internal energy, it is shown that these models have a third order phase transition.

Loop spaces of the Q-construction

Giffen in [1], and Gillet-Grayson in [3], independently found a simplicial model for the loop space on Quillen's Q-construction. Their proofs work for exact categories. Here we generalise the results to the K-theory of triangulated categories. The old pro

Motivic symmetric spectra

This paper demonstrates the existence of a theory of symmetric spectra for the motivic stable category. The main results together provide a categorical model for the motivic stable category which has an internal symmetric monoidal smash product. The details of the basic construction of the Morel-Voevodsky proper closed simplicial model structure underlying the motivic stable category are required to handle the symmetric case, and are displayed in the first three sections of this paper.

The Area of the Medial Parallelogram of a Tetrahedron

(1999). The Area of the Medial Parallelogram of a Tetrahedron. The American Mathematical Monthly: Vol. 106, No. 10, pp. 956-958.

Anisotropic simplicial minisuperspace model in the presence of a scalar field

We study an anisotropic simplicial minisuperspace model with a cosmological constant and a massive scalar field. After obtaining the classical solutions we then compute the semiclassical approximation of the no-boundary wavefunction of the Universe along a steepest-descent contour passing through classical Lorentzian solutions. The oscillatory behaviour of the resulting wavefunction is consistent with the prediction of classical Lorentzian spacetime for the late Universe.