Related papers: Comparison theorems for Kan, faintly universal and…
We give a model-independent construction of directed univalent cocartesian fibrations of $(\infty,1)$-categories, and prove a straightening equivalence against such fibrations. The key step is showing that cocartesian fibrations descend…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
We study the structure of the category of representations of $\mathbf{FA}$, the category of finite sets and all maps, mostly working over a field of characteristic zero. This category is not semi-simple and exhibits interesting features. We…
We establish a Dwyer-Kan equivalence of relative categories of combinatorial model categories, presentable quasicategories, and other models for locally presentable (infinity,1)-categories. This implies that the underlying quasicategories…
We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the…
Given a functor $T:C \to D$ carrying a class of morphisms $S\subset C$ into a class $S'\subset D$, we give sufficient conditions in order that $T$ induces an equivalence on the localised categories. These conditions are in the spirit of…
We provide an $(\infty,n)$-categorical version of the straightening-unstraightening construction, asserting an equivalence between the $(\infty,n)$-category of double $(\infty,n-1)$-right fibrations over an $(\infty,n)$-category…
Given a bounding class $B$, we construct a bounded refinement $BK(-)$ of Quillen's $K$-theory functor from rings to spaces. $BK(-)$ is a functor from weighted rings to spaces, and is equipped with a comparison map $BK \to K$ induced by…
We introduce and study a general notion of polynomial functor from a small monoidal symmetric category whose unit is an initial object and give a classification result of polynomial functors of degree smaller of equal to n modulo those of…
Let $\mathcal{Z}$ be a specialization closed subset of $\Spec R$ and $X$ a homologically left-bounded complex with finitely generated homologies. We establish Faltings' Local-global Principle and Annihilator Theorems for the local…
We study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Marki and Tholen's semi-abelian categories. This model structure exists as soon as the base category A is regular Mal'tsev and has…
A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a…
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can…
Homotopical localizations with respect to (possibly proper) classes of maps are known to exist assuming the validity of a large-cardinal axiom from set theory called Vop\v{e}nka's principle. In this article, we prove that each of the…
We introduce the notion of an effective Kan fibration, a new mathematical structure that can be used to study simplicial homotopy theory. Our main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective…
We specialise a recently introduced notion of generalised dinaturality for functors $T : (\mathcal{C}^\text{op})^p \times \mathcal{C}^q \to \mathcal{D}$ to the case where the domain (resp., codomain) is constant, obtaining notions of ends…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through *-autonomous monoidal categories and related structures.
Coherent strings of composable morphisms play an important role in various important constructions in abstract stable homotopy theory (for example algebraic K-theory or higher Toda brackets) and in the representation theory of finite…
We show that Martin Hyland's effective topos can be exhibited as the homotopy category of a path category $\mathbb{EFF}$. Path categories are categories of fibrant objects in the sense of Brown satisfying two additional properties and as…