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We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial…

Category Theory · Mathematics 2016-09-16 Simon Henry

We define an equivalence of categories, quite similar to the classical Cartier-Milnor-Moore-Quillen theorem, between the category of connected dendriform bialgebras and the category of brace algebras. This equivalence is given by the…

Quantum Algebra · Mathematics 2007-05-23 Frederic Chapoton

We introduce, for \(\C\) a regular Cartesian Reedy category a model category whose fibrant objects are an analogue of quasicategories enriched in simplicial presheaves on \(C\). We then develop a coherent realization and nerve for this…

Category Theory · Mathematics 2019-10-01 Harry Gindi

For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a…

Category Theory · Mathematics 2021-08-16 Nima Rasekh

We introduce a relative version of the $2$-Segal simplicial spaces defined by Dyckerhoff and Kapranov and G\'{a}lvez-Carrillo, Kock and Tonks. Examples of relative $2$-Segal spaces include the categorified unoriented cyclic nerve, real…

Representation Theory · Mathematics 2018-03-16 Matthew B. Young

To any affine scheme with a $\mathbb{G}_m$-action, we provide a Bousfield colocalization on the equivariant derived category of modules by constructing, via homotopical methods, an idempotent integral kernel. This endows the equivariant…

Algebraic Geometry · Mathematics 2017-10-05 Matthew R. Ballard , Colin Diemer , David Favero

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to…

Category Theory · Mathematics 2010-04-21 Michael Batanin , Denis-Charles Cisinski , Mark Weber

Let $X$ be a variety with an action by an algebraic group $G$. In this paper we discuss various properties of $G$-equivariant $D$-modules on $X$, such as the decompositions of their global sections as representations of $G$ (when $G$ is…

Algebraic Geometry · Mathematics 2019-04-11 András C. Lőrincz , Uli Walther

We use the terms "$\infty$-categories" and "$\infty$-functors" to mean the objects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal…

Category Theory · Mathematics 2019-09-23 Emily Riehl , Dominic Verity

We associate a t-structure to a family of objects in D(A), the derived category of a Grothendieck category A. Using general results on t-structures, we give a new proof of Rickard's theorem on equivalence of bounded derived categories of…

Representation Theory · Mathematics 2007-05-23 Leovigildo Alonso , Ana Jeremias , Ma. -Jose Souto

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we…

Category Theory · Mathematics 2021-08-19 Joe Moeller , Christina Vasilakopoulou

Evidence is given for the correctness of the Joyal-Riehl-Verity construction of the homotopy bicategory of the $(\infty, 2)$-category of $(\infty, 1)$-categories; in particular, it is shown that the analogous construction using complete…

Category Theory · Mathematics 2013-11-05 Zhen Lin Low

We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit small and practical algebraic model, thereby providing a universal de Rham model for rational G-equivariant cohomology theories. The result…

Algebraic Topology · Mathematics 2018-07-04 J. P. C. Greenlees , B. Shipley

We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $\Omega$-$G$-spectra for any topological…

Algebraic Topology · Mathematics 2025-08-04 J. Peter May , Mona Merling , Angélica M. Osorno

We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be…

Rings and Algebras · Mathematics 2015-04-07 Eva Bayer-Fluckiger , Uriya A. First , Daniel A. Moldovan

We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of…

Quantum Algebra · Mathematics 2008-02-08 Martin Andler , Siddhartha Sahi

The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with…

Differential Geometry · Mathematics 2020-06-03 Aleksander Doan , Thomas Walpuski

This thesis is divided into two parts. In the first part, we study models of $(\infty,\omega)$-categories. The main result is to establish a Quillen equivalence between Rezk's complete Segal $\Theta$-spaces and Verity's complicial sets. In…

Category Theory · Mathematics 2023-10-13 Félix Loubaton

Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of…

Algebraic Geometry · Mathematics 2015-05-13 Alexei Elagin

Using Dugger's construction of universal model categories, we produce replacements for simplicial and combinatorial symmetric monoidal model categories with better operadic properties. Namely, these replacements admit a model structure on…

Algebraic Topology · Mathematics 2024-12-31 Haldun Özgür Bayındır , Boris Chorny