Bar Construction Research Articles

Action of Crossed Modules and Bar Construction

If a group $N$ acts on a set $X$, a simplicial set $Bar(X,N)$ using the usual bar construction has been provided. In this construction, if the group $N$ acts on a group $G$ via a homomorphism $f:N\rightarrow G$, then $Bar(G,N)$ has a simplicial set structure. In the case of $f$ has a crossed module structure, $Bar(G,N)$ has a normal simplicial group structure. In this work, by defining an action of a crossed module $\partial: N_1 \longrightarrow X_1$ on a homomorphism of groups $f: N_2 \longrightarrow X_2 $ via a double map $\alpha: \partial\rightarrow f$, we will construct a bisimplicial set, using the 2-dimensional version of the usual Bar construction.

Rates of Evacuation of Bedload Sediment From an Alpine Glacier Control Proglacial Stream Morphodynamics

AbstractProglacial forefields commonly include highly dynamic fluvial systems associated with the fundamental instability between topography, flow hydraulics and sediment transport. However, there is limited knowledge of how these systems respond to changing subglacial hydrology and sediment supply. We investigated this relationship using the first continuous field‐collected data sets for both suspended and bedload sediment export and proglacial river dynamics for an Alpine glacier forefield, the Glacier d’Otemma, Switzerland. The results show a strong sensitivity of fluvial morphodynamics to the balance between sediment transport capacity and supply. When subglacial bedload export rates exceeded fluvial transport capacity, we found bar construction leading to net forefield aggradation and surficial coarsening, especially on bar heads. This intensified braiding buffered the downstream transport of coarse sediment. When subglacial bedload export rates were lower than transport capacity, incision occurred, with reduced braiding intensity, net erosion and important amounts of bedload leaving the proglacial system. We found a net fining of surficial deposits except for very isolated coarsening patterns on bar heads. Thus, proglacial forefield morphodynamics are strongly conditioned by subglacial hydrology and sediment supply, but this conditioning is also influenced by the response of the forefield itself. Proglacial forefields have an important influence on the longitudinal connectivity of sediment flux in regions sensitive to climate change, such as recently deglaciated high mountain areas. The linkages we report between subglacial processes and river morphodynamics are critical for understanding the development of embryonic forefield ecosystems.

Numerical Solution of Natural Frequencies and Mode Shapes

Every construction has a set of their natural frequencies (resonant frequencies) and mode shapes that depend on its material, structure, and boundary conditions. A mode shape is a deflection pattern. It is related to a particular natural frequency and represents the relative displacement of all parts of a construction for that particular mode. This paper deals with numerical solutions of natural frequencies and mode shapes. The method of stiffness constants and bisection method is used. The procedure can be used for any planar bar construction. First, the mode shapes of the simple beam are solved, where the solution is compared with the results obtained using known relationships. The article further describes the solution of the parabolic arc.

A geometric approach to equivariant factorization homology and nonabelian Poincaré duality

Fix a finite group G and an n-dimensional orthogonal G-representation V. We define the equivariant factorization homology of a V-framed smooth G-manifold with coefficients in an $$\textrm{E}_V$$ -algebra using a two-sided bar construction, generalizing (Andrade, From manifolds to invariants of $$E_n$$ -algebras. PhD thesis, Massachusetts Institute of Technology, 2010; Kupers and Miller, Math Ann 370(1–2):209–269, 2018). This construction uses minimal categorical background and aims for maximal concreteness, allowing convenient proofs of key properties, including invariance of equivariant factorization homology under change of tangential structures. Using a geometrically-seen scanning map, we prove an equivariant version (eNPD) of the nonabelian Poincaré duality theorem due to several authors. The eNPD states that the scanning map gives a G-equivalence from the equivariant factorization homology to mapping spaces out the one-point compactification of the G-manifolds, when the coefficients are G-connected. For non-G-connected coefficients, when the G-manifolds have suitable copies of $$\mathbb {R}$$ in them, the scanning map gives group completions. This generalizes the recognition principle for V-fold loop spaces in Guillou and May (Algebr Geom Topol 17(6):3259–3339, 2017).

Plethysms and operads

We introduce the T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document}-construction, an endofunctor on the category of generalized operads, as a general mechanism by which various notions of plethystic substitution arise from more ordinary notions of substitution. In the special case of one-object unary operads, i.e. monoids, we recover the T-construction of Giraudo. We realize several kinds of plethysm as convolution products arising from the homotopy cardinality of the incidence bialgebra of the bar construction of various operads obtained from the T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document}-construction. The bar constructions are simplicial groupoids, and in the special case of the terminal reduced operad Sym\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {Sym}$$\\end{document}, we recover the simplicial groupoid of Cebrian (Algebraic Geom Topol 21(1):421–446, 2021), a combinatorial model for ordinary plethysm in the sense of Pólya, given in the spirit of Waldhausen S and Quillen Q constructions. In some of the cases of the T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document}-construction, an analogous interpretation is possible.

Modeling bundle-valued forms on the path space with a curved iterated integral

The usual iterated integral map given by Chen produces an equivalence between the two-sided bar complex on differential forms and the de Rham complex on the path space. This map fails to make sense when considering the curved differential graded algebra of bundle-valued forms with a covariant derivative induced by a connection. In this paper, we define a curved version of Chen’s iterated integral that incorporates parallel transport and maps an analog of the two-sided bar construction on bundle-valued forms to bundle-valued forms on the path space. This iterated integral is proven to be a homotopy equivalence of curved differential graded algebras, and for real-valued forms it factors through the usual Chen iterated integral.

Algebraic 𝐾-theory of 𝑇𝐻𝐻(𝔽_{𝕡})

In this work we study the E ∞ E_{\infty } -ring THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) as a graded spectrum. Following an identification at the level of E 2 E_2 -algebras with F p [ Ω S 3 ] \mathbb {F}_p[\Omega S^3] , the group ring of the E 1 E_1 -group Ω S 3 \Omega S^3 over F p \mathbb {F}_p , we show that the grading on THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) arises from decomposition on the cyclic bar construction of the pointed monoid Ω S 3 \Omega S^3 . This allows us to use trace methods to compute the algebraic K K -theory of THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) . We also show that as an E 2 E_2 H F p H\mathbb {F}_p -ring, THH ⁡ ( F p ) \operatorname {THH}(\mathbb {F}_p) is uniquely determined by its homotopy groups. These results hold in fact for THH ⁡ ( k ) \operatorname {THH}(k) , where k k is any perfect field of characteristic p p . Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic K K -theory of formal DGAs.

Kan Extensions are Partial Colimits

One way of interpreting a left Kan extension is as taking a kind of “partial colimit”, whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the partial evaluations sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the restriction-of-scalars construction of monads extends to the case of pseudoalgebras over pseudomonads, we consider a morphism of monads between them, which we call image. This morphism allows in particular to generalize the idea of confinal functors, i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.

Hyperoctahedral homology for involutive algebras

Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group.

Koszul duality for compactly generated derived categories of second kind

For any dg algebra A we construct a closed model category structure on dg A -modules such that the corresponding homotopy category is compactly generated by dg A -modules that are finitely generated and free over A (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a so-called extended bar construction of A . This generalises and complements certain aspects of dg Koszul duality for associative algebras.

The derived deformation theory of a point

We provide a prorepresenting object for the noncommutative derived deformation problem of deforming a module $X$ over a differential graded algebra. Roughly, we show that the corresponding deformation functor is homotopy prorepresented by the dual bar construction on the derived endomorphism algebra of $X$. We specialise to the case when $X$ is one-dimensional over the base field, and introduce the notion of framed deformations, which rigidify the problem slightly and allow us to obtain derived analogues of the results of Ed Segal's thesis. Our main technical tool is Koszul duality, following Pridham and Lurie's interpretation of derived deformation theory. Along the way we prove that a large class of dgas are quasi-isomorphic to their Koszul double dual, which we interpret as a derived completion functor; this improves a theorem of Lu-Palmieri-Wu-Zhang. We also adapt our results to the setting of multi-pointed deformation theory, and furthermore give an analysis of universal prodeformations. As an application, we give a deformation-theoretic interpretation to Braun-Chuang-Lazarev's derived quotient.

Using the Particle Model to Find Structure in Eilenberg-MacLane Spaces

We introduce two models, the bar construction and the particle model, for a collection of spaces known as Eilenberg-MacLane spaces. These spaces can be used as building blocks for other spaces and are of great interest throughout topology. We use these models to compute the homology groups for one Eilenberg-MacLane space in an intuitive and accessible way, and then show how the models can be used to determine the cell structure for additional cases using combinatorial arguments. An introduction to cellular homology and examples are included along the way.

Solution to the Problem of Steel Bar Construction of Semi-Inverted T-shaped Invisible Cover Beam Based on BIM Technology

Solution to the Problem of Steel Bar Construction of Semi-Inverted T-shaped Invisible Cover Beam Based on BIM Technology

Homotopy Gerstenhaber formality of Davis–Januszkiewicz spaces

A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. We prove that the normalized singular cochain algebra of a Davis-Januszkiewicz space is formal as a homotopy Gerstenhaber algebra, for any coefficient ring. This generalizes a recent result by the author about classifying spaces of tori and also strengthens the well-known dga formality result for Davis-Januszkiewicz spaces due to the author and Notbohm-Ray. As an application, we determine the cohomology rings of free and based loop spaces of Davis-Januszkiewicz spaces.

ℰn–Hopf invariants

The classical Hopf invariant is an invariant of homotopy classes of maps from $S^{4n-1} $ to $S^{2n}$, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for $E_n$-operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space $X$ and the homotopy groups of $X$. In this paper we will give a generalization of the classical Hopf invariant by defining a pairing between the cohomology of the $E_n$-bar construction on the cochains of $X$ and the homotopy groups of $X$. This pairing gives us a set of invariants of homotopy classes of maps from $S^m$ to a simplicial set $X$, this pairing can detect more homotopy classes of maps than the classical Hopf invariant. The second part of the paper is devoted to combining the $E_n$-Hopf invariants with the Koszul duality theory for $E_n$-operads to get a relation between the $E_n$-Hopf invariants of a space $X$ and the $E_{n+1}$-Hopf invariants of the suspension of $X$. This is done by studying the suspension morphism for the $E_\infty$-operad, which is a morphism from the $E_{\infty}$-operad to the desuspension of the $E_\infty$-operad. We show that it induces a functor from $E_\infty$-algebras to $E_\infty$-algebras, which has the property that it sends an $E_\infty$-model for a simplicial set $X$ to an $E_\infty$-model for the suspension of $X$. We use this result to give a relation between the $E_n$-Hopf invariants of maps from $S^m$ into $X$ and the $E_{n+1}$-Hopf invariants of maps from $S^{m+1}$ into the suspension of $X$. One of the main results we show here, is that this relation can be used to define invariants of stable homotopy classes of maps.

Higher dimensional algebras as ideal maps

In this work, we explain the close relationship between an ideal map structure $S\rightarrow End_{R}(R)$ on a homomorphism of commutative $k$-algebras $R\rightarrow S$ and an ideal simplicial algebra structure on the associated bar construction $Bar(S,R)$. We also explain this structure for crossed squares of algebras.

Holomorphic bundles on the blown-up plane and the bar construction

We study the moduli space $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ of rank $r$ holomorphic bundles with trivial determinant and second Chern class $c_2=k$, over the blowup $\tilde{\mathbb P}^2_{\!q}$ of the projective plane at $q$ points, trivialized on a rational curve. We show that, for $k=1,2$, we have a homotopy equivalence between $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ and the degree $k$ component of the bar construction $\mathrm{B}\bigl(\mathfrak M^r\mathbb P^2,(\mathfrak M^r\mathbb P^2)^{q},(\mathfrak M^r\tilde{\mathbb P}_{\!1}^2)^{q}\bigr)$. The space $\mathfrak M_k^r(\tilde{\mathbb P}^2_{\!q})$ is isomorphic to the moduli space $\mathfrak M\mathcal I_k^r(X_q)$ of charge $k$ based $SU(r)$ instantons on a connected sum $X_q$ of $q$ copies of $\overline{\mathbb P^2}$ and we show that, for $k=1,2$, we have a homotopy equivalence between $\mathfrak M\mathcal I_k^r(X_q\# X_s)$ and the degree $k$ component of $\mathrm{B}\bigl(\mathfrak M\mathcal I^r(X_q),\mathfrak M\mathcal I^r(S^4),\mathfrak M\mathcal I^r(X_s)\bigr)$. Analogous results hold in the limit when $k\to\infty$. As an application we obtain upper bounds for the cokernel of the Atiyah-Jones map in homology, in the rank-stable limit.

Homotopy Gerstenhaber algebras are strongly homotopy commutative

We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff-Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a cup-1 product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense of Munkholm.

Monads, Partial Evaluations, and Rewriting

Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of “evaluating an expression partially”: for example, “2+3” can be obtained as a partial evaluation of “2+2+1”. This construction can be given for any monad, and it is linked to the famous bar construction [Saunders Mac Lane, Categories for the Working Mathematician, Springer, 2000, VII.6], of which it gives an operational interpretation: the bar construction is a simplicial set, and its 1-cells are partial evaluations.We study the properties of partial evaluations for general monads. We prove that whenever the monad is weakly cartesian, partial evaluations can be composed via the usual Kan filler property of simplicial sets, of which we give an interpretation in terms of substitution of terms.For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables, and partial evaluation relation is known as second-order stochastic dominance.In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction.

Diagram Complexes, Formality, and Configuration Space Integrals for Spaces of Braids

AbstractWe use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen’s iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a commutative differential graded algebra suitable for integration. We show that this integration is compatible with Bott–Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.