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We use the notion of multi-Reedy category to prove that, if $\mathcal C$ is a Reedy category, then $\Theta \mathcal C$ is also a Reedy category. This result gives a new proof that the categories $\Theta_n$ are Reedy categories. We then…

Algebraic Topology · Mathematics 2012-12-20 Julia E. Bergner , Charles Rezk

Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a…

Logic · Mathematics 2022-06-10 Alec Rhea

We describe a comparison between pretriangulated differential graded categories and certain stable infinity categories. Specifically, we use a model category structure on differential graded categories over k (a field of characteristic 0)…

Algebraic Topology · Mathematics 2016-09-13 Lee Cohn

If ${\cal D}$ is a definable category then it may contain no nonzero finitely presented modules but, by a result of Makkai, there is a $\varinjlim$-generating set of strictly ${\cal D}$-atomic modules. These modules share some key…

Representation Theory · Mathematics 2024-02-09 Mike Prest

Tate's central extension originates from 1968 and has since found many applications to curves. In the 80s Beilinson found an n-dimensional generalization: cubically decomposed algebras, based on ideals of bounded and discrete operators in…

Algebraic Geometry · Mathematics 2017-03-31 Oliver Braunling , Michael Groechenig , Jesse Wolfson

Based on the intuitive notion of convexity, we formulate a universal property defining interval objects in a category with finite products. Interval objects are structures corresponding to closed intervals of the real line, but their…

Category Theory · Mathematics 2025-05-01 Martin Escardo , Alex Simpson

Object Permanence allows people to reason about the location of non-visible objects, by understanding that they continue to exist even when not perceived directly. Object Permanence is critical for building a model of the world, since…

Computer Vision and Pattern Recognition · Computer Science 2020-07-17 Aviv Shamsian , Ofri Kleinfeld , Amir Globerson , Gal Chechik

We shall show that the stable categories of graded Cohen-Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our…

Representation Theory · Mathematics 2011-02-17 Osamu Iyama , Ryo Takahashi

We introduce new genuine zetas. There are two types, i.e., the pure non- abelian zetas defined using semi-stable bundles, and the group zetas defined for reductive groups. Basic properties such as rationality and functional equation are…

Algebraic Geometry · Mathematics 2012-02-21 Lin Weng

The main goal of this paper is to generalize Serre-Tate theory of "ordinary" local moduli to Shimura varieties of PEL type. To this end we develop a generalized notion of ordinariness, we prove a number of basic results about this, and we…

Algebraic Geometry · Mathematics 2007-05-23 Ben Moonen

We find new families of shape invariant potentials depending on n>=1 parameters subject to translation by the inclusion of non-trivial invariants. New dependencies of the spectra are found, and it opens the door to the engineering of…

Quantum Physics · Physics 2022-05-11 Arturo Ramos

We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an…

Differential Geometry · Mathematics 2012-01-30 Thomas Leuther

In many everyday categories (sets, spaces, modules, ...) objects can be both added and multiplied. The arithmetic of such objects is a challenge because there is usually no subtraction. We prove a family of cases of the following principle:…

Category Theory · Mathematics 2010-02-04 Marcelo Fiore , Tom Leinster

TDA (topological data analysis) is a relatively new area of research related to importing classical ideas from topology into the realm of data analysis. Under the umbrella term TDA, there falls, in particular, the notion of persistent…

Algebraic Topology · Mathematics 2019-06-03 Facundo Memoli , Kritika Singhal

We identify a class of symmetric algebras over a complete discrete valuation ring $\mathcal O$ of characteristic zero to which the characterisation of Kn\"orr lattices in terms of stable endomorphism rings in the case of finite group…

Representation Theory · Mathematics 2018-03-16 Florian Eisele , Michael Geline , Radha Kessar , Markus Linckelmann

For a tensor triangulated category and any regular cardinal $\alpha$ we study the frame of $\alpha$-localizing tensor ideals and its associated space of points. For a well-generated category and its frame of localizing tensor ideals we…

Category Theory · Mathematics 2022-09-07 Henning Krause , Janina C. Letz

We propose a construction of a stable category for any pretorsion theory in a lextensive category. We prove the universal property of the stable category, that extends previous results obtained for the stable category of internal preorders…

Category Theory · Mathematics 2023-01-16 Francis Borceux , Federico Campanini , Marino Gran

I study definable sets in affine continuous logic. Let $T$ be an affine theory. After giving some general results, it is proved that if $T$ has a first order model, its extremal theory is a complete first order theory and first order…

Logic · Mathematics 2024-03-13 Seyed-Mohammad Bagheri

We define natural A_infinity-transformations and construct A_infinity-category of A_infinity-functors. The notion of non-strict units in an A_infinity-category is introduced. The 2-category of (unital) A_infinity-categories, (unital)…

Category Theory · Mathematics 2008-02-17 Volodymyr Lyubashenko

We prove that the number of parameters defining a complex of projective modules over a finite dimensional algebra is upper semi-continuous in families of algebras. Supposing that every algebra is either derived tame or derived wild, we get…

Representation Theory · Mathematics 2007-05-23 Yuriy A. Drozd