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Register a new userA computer algebra system for the study of commutativity up-to-coherent homotopies
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Added by
Anibal M. Medina-Mardones
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Anibal M. Medina-Mardones
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The Python package ComCH is a lightweight specialized computer algebra system that provides models for well known objects, the surjection and Barratt-Eccles operads, parameterizing the product structure of algebras that are commutative in a derived sense. The primary examples of such algebras treated by ComCH are the cochain complexes of spaces, for which it provides effective constructions of Steenrod cohomology operations at all prime.
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In research problems that involve the use of numerical methods for solving systems of ordinary differential equations (ODEs), it is often required to select the most efficient method for a particular problem. To solve a Cauchy problem for a system of ODEs, Runge-Kutta methods (explicit or implicit ones, with or without step-size control, etc.) are employed. In that case, it is required to search through many implementations of the numerical method and select coefficients or other parameters of its numerical scheme. This paper proposes a library and scripts for automated generation of routine functions in the Julia programming language for a set of numerical schemes of Runge-Kutta methods. For symbolic manipulations, we use a template substitution tool. The proposed approach to automated generation of program code allows us to use a single template for editing, instead of modifying each individual function to be compared. On the one hand, this provides universality in the implementation of a numerical scheme and, on the other hand, makes it possible to minimize the number of errors in the process of modifying the compared implementations of the numerical method. We consider Runge-Kutta methods without step-size control, embedded methods with step-size control, and Rosenbrock methods with step-size control. The program codes for the numerical schemes, which are generated automatically using the proposed library, are tested by numerical solution of several well-known problems.
Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an algebraic pattern, bywhich we mean an $infty$-category equipped with a factorization system and a collection of elementary objects. Examples of structures that occur as such Segal $mathcal{O}$-spaces for an algebraic pattern $mathcal{O}$ include $infty$-categories, $(infty,n)$-categories, $infty$-operads, $infty$-properads, and algebras for an $infty$-operad in spaces. In the first part of this paper we set up a general frameworkn for algebraic patterns and their Segal objects, including conditions under which the latter are preserved by left and right Kan extensions. In particular, we obtain necessary and sufficent conditions on a pattern $mathcal{O}$ for free Segal $mathcal{O}$-spaces to be described by an explicit colimit formula, in which case we say that $mathcal{O}$ is extendable. In the second part of the paper we explore the relationship between extendable algebraic patterns and polynomial monads, by which we mean cartesian monads on presheaf $infty$-categories that are accessible and preserve weakly contractible limits. We first show that the free Segal $mathcal{O}$-space monad for an extendable pattern $mathcal{O}$ is always polynomial. Next, we prove an $infty$-categorical version of Webers Nerve Theorem for polynomial monads, and use this to define a canonical extendable pattern from any polynomial monad, whose Segal spaces are equivalent to the algebras of the monad. These constructions yield functors between polynomial monads and extendable algebraic patterns, and we show that these exhibit full subcategories of saturated algebraic patterns and complete polynomial monads as localizations, and moreover restrict to an equivalence between the $infty$-categories of saturated patterns and complete polynomial monads.
The notion of a derived A-infinity algebra, considered by Sagave, is a generalization of the classical notion of A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r-homotopy, for non-negative integers r, in such a way that r-homotopy equivalences underlie E_r-quasi-isomorphisms, defined via an associated spectral sequence. We study the special case of twisted complexes (also known as multicomplexes) first since it is of independent interest and this simpler case clearly exemplifies the structure we study. We also give two new interpretations of derived A-infinity algebras as A-infinity algebras in twisted complexes and as A-infinity algebras in split filtered cochain complexes.
We study the mod-$ell$ homotopy type of classifying spaces for commutativity, $B(mathbb{Z}, G)$, at a prime $ell$. We show that the mod-$ell$ homology of $B(mathbb{Z}, G)$ depends on the mod-$ell$ homotopy type of $BG$ when $G$ is a compact connected Lie group, in the sense that a mod-$ell$ homology isomorphism $BG to BH$ for such groups induces a mod-$ell$ homology isomorphism $B(mathbb{Z}, G) to B(mathbb{Z}, H)$. In order to prove this result, we study a presentation of $B(mathbb{Z}, G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an idea of Adem and Gomez. We also study the relationship between the mod-$ell$ type of a Lie group $G(mathbb{C})$ and the locally finite group $G(bar{mathbb{F}}_p)$ where $G$ is a Chevalley group. We see that the naive analogue for $B(mathbb{Z}, G)$ of the celebrated Friedlander--Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a $G$ action on $B(mathbb{Z}, G)$.
The commutative differential graded algebra $A_{mathrm{PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{mathcal{I}}(X)$ of $A_{mathrm{PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $mathcal{I}$ to model $E_{infty}$ differential graded algebras by strictly commutative objects, called commutative $mathcal{I}$-dgas. We define a functor $A^{mathcal{I}}$ from simplicial sets to commutative $mathcal{I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{infty}$ dga of cochains. The functor $A^{mathcal{I}}$ shares many properties of $A_{mathrm{PL}}$, and can be viewed as a generalization of $A_{mathrm{PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{mathcal{I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.
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