Infinity

Summary

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol \infty. Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes.

Official source

https://en.wikipedia.org/wiki/Infinity

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In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notion of a Hopf algebra H in an E-2-monoidal infinity-category and lift some results about ordinary Hopf algebras, such as the fundamental theorem of Hopf modules, to this setting. We also study Hopf-Galois extensions in this context. Given a candidate Hopf-Galois extension, i.e., a map f : A -> B of H-comodule algebras where H coacts on A trivially, we construct a structured version of the comparison map B (x)_A B -> H (x) B that allows us to compare the category of descent data for f with a category of "B-modules equipped with a semilinear coaction of H". We provide further insights into the case of commutative (i.e., E-infinity) comodule algebras over a commutative Hopf algebra, for instance a description of the aforementioned category of modules equipped with a semilinear coaction as the limit of a "categorified cobar construction". Moreover, we provide a simple description of comodules over a space in slice categories of the infinity-category of spaces, which enables us to realize multiplicative Thom objects as comodule algebras and thus incorporate them into the aforementioned framework.

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Homotopical relations between 2-dimensional categories and their infinity-analogues

Lyne Moser

In this thesis, we study the homotopical relations of 2-categories, double categories, and their infinity-analogues. For this, we construct homotopy theories for the objects of interest, and show that there are homotopically full embeddings of 2-categories into (infinity,2)-categories, and of double categories into double (infinity,1)-categories, which are compatible with the inclusions of 2-categories and (infinity,2)-categories into their double categorical analogues. In the strict setting, we first present two model structures on the category of double categories and double functors, constructed in papers by the author, Sarazola, and Verdugo. Unlike previously defined model structures for double categories, they recover Lack's model structure for 2-categories. More precisely, the horizontal embedding functor from 2-categories to double categories is homotopically well-behaved, and embeds the homotopy theory of 2-categories into that of double categories in a reflective way. While, in the first model structure, all double categories are fibrant, the fibrant objects of the second model structure are the weakly horizontally invariant double categories. We show that both model structures are enriched over 2-categories, and that the model structure for weakly horizontally invariant double categories is further monoidal with respect to the Gray tensor product for double categories. Going to the infinity-world, we then consider infinity-versions of these 2-dimensional categories. Double (infinity,1)-categories are defined as double Segal objects in spaces which are complete in the horizontal direction, and hence include (infinity,2)-categories in the form of Barwick's 2-fold complete Segal spaces. We then construct a nerve from double categories to double (infinity,1)-categories, and show that it embeds the homotopy theory for weakly horizontally invariant double categories into that of double (infinity,1)-categories in a reflective way. Finally, by restricting the nerve along the horizontal embedding, we obtain a nerve from 2-categories to 2-fold complete Segal spaces, which again embeds the homotopy theory of 2-categories into that of (infinity,2)-categories in a reflective way.

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We consider the parabolic Anderson model driven by fractional noise: partial derivative/partial derivative t u(t,x) = k Delta u(t,x) + u(t,x)partial derivative/partial derivative t W(t,x) x is an element of Z(d), t >= 0, where k > 0 is a diffusion constant, Delta is the discrete Laplacian defined by Delta f (x) = 1/2d Sigma vertical bar y-x vertical bar=1 (f(Y) - f (0)), and {W(t,x) ; t > 0} x is an element of Z(d) is a family of independent fractional Brownian motions with Hurst parameter H is an element of (0,1), indexed by Z(d). We make sense of this equation via a Stratonovich integration obtained by approximating the fractional Brownian motions with a family of Gaussian processes possessing absolutely continuous sample paths. We prove that the Feynman-Kac representation u(t, x) = E-infinity [u(0) (X(t)) exp integral(t)(0) W(ds,X(t-s))], (1) is a mild solution to this problem. Here u(0) (y) is the initial value at site y is an element of Z(d), {X(t); t >= 0} is a simple random walk with jump rate f, started at x E Zd and independent of the family {W(t, x) ; t >= 0}(x is an element of Zd) and E-infinity is expectation withrespect to this random walk. We give a unified argument that works for any Hurst parameter H is an element of (0,1). (C) 2015 Elsevier Inc. All rights reserved.

Academic Press Inc Elsevier Science2015

Homotopical relations between 2-dimensional categories and their infinity-analogues

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EPFL2021