Quillen's Theorem Research Articles

A resolution theorem for extriangulated categories with applications to the index

Quillen's Resolution Theorem in algebraic K-theory provides a powerful computational tool for calculating K-groups of exact categories. At the level of K0, this result goes back to Grothendieck. In this article, we first establish an extriangulated version of Grothendieck's Resolution Theorem.Second, we use this Extriangulated Resolution Theorem to gain new insight into the index theory of triangulated categories. Indeed, we propose an index with respect to an extension-closed subcategory N of a triangulated category C and we prove an additivity formula with error term. Our index recovers the index with respect to a contravariantly finite, rigid subcategory X defined by Jørgensen and the second author, as well as an isomorphism between K0sp(X) and the Grothendieck group of a relative extriangulated structure CRX on C when X is n-cluster tilting. In addition, we generalize and enhance some results of Fedele. Our perspective allows us to remove certain restrictions and simplify some arguments.Third, as another application of our ExtriangulatedResolution Theorem, we show that if X is n-cluster tilting in an abelian category, then the index introduced by Reid gives an isomorphism K0(CRX)≅K0sp(X).

Theorem A for marked 2-categories

In this work, we prove a generalization of Quillen's Theorem A to 2-categories equipped with a special set of morphisms which we think of as weak equivalences, providing sufficient conditions for a 2-functor to induce an equivalence on ( ∞ , 1 ) -localizations. When restricted to 1-categories with all morphisms marked, our theorem retrieves the classical Theorem A of Quillen. We additionally state and provide evidence for a new conjecture: the cofinality conjecture , which describes the relation between a conjectural theory of marked ( ∞ , 2 ) -colimits and our generalization of Theorem A.

2-Final 2-functors

A final functor between categories F:A→B is a functor that allows the restriction of diagrams on B to A without changing their colimits. More precisely, the functor F is final if, for any diagram D:B→E, there is a canonical isomorphismcolimBD≅colimAD∘F where either colimit exists whenever the other one does. There is a classical criterion for final functors [6, §IX.3]: a functor F:A→B is final if and only if, for any object b∈B, the slice category b/F is nonempty and connected. Such a criterion also exists for (∞,1)-categories, under the name of Quillen's Theorem A [5, §4.1]: an (∞,1)-functor F:A→B is final (with respect to any ∞-diagram) if and only if for any object b∈B, the slice ∞-category b/F is weakly contractible. One would expect a similar result for any dimension: an n-functor F:A→B is final (with respect to any n-diagram) if and only if, for any object b∈B, the slice n-category b/F is nonempty and has trivial homotopy groups πk for 0≤k≤n−1. Note that these are not consequences of the known criterion for 1-functors and (∞,1)-functors (see Remark 2.11 and Remark 3.3). A related result for 2-filtered 2-functors is proved in [1, §1.3].This paper presents a combinatorial proof in the case n=2 (Theorem 3.4). A general application is given in section 4, which can be specialized to 2-categories relevant to the study of finite groups.

Lannes’s T–functor and equivariant Chow rings

For $X$ a smooth scheme acted on by a linear algebraic group $G$ and $p$ a prime, the equivariant Chow ring $CH^*_G(X)\otimes \mathbb{F}_p$ is an unstable algebra over the Steenrod algebra. We compute Lannes's $T$-functor applied to $CH^*_G(X)\otimes \mathbb{F}_p$. As an application, we compute the localization of $CH^*_G(X)\otimes \mathbb{F}_p$ away from $n$-nilpotent modules over the Steenrod algebra, affirming a conjecture of Totaro as a special case. The case when $X$ is a point and $n = 1$ generalizes and recovers an algebro-geometric version of Quillen's stratification theorem proved by Yagita and Totaro.

Simple homotopy theory and nerve theorem for categories

We study a combinatorial homotopy theory for small categories without a loop (loopfree categories) which is closely related to Whitehead's simple homotopy theory for regular CW-complexes with triangular cells. Quillen's theorem A and the nerve theorem for loopfree categories are considered from the viewpoint of the simple homotopy theory. Moreover, we extend the classical nerve theorem and discuss an application to the topological data analysis for data-sets lying in a non-convex field.

Homology of categories via polygraphic resolutions

In this paper, we prove that the polygraphic homology of a small category, defined in terms of polygraphic resolutions in the category ω Cat of strict ω - categories, is naturally isomorphic to the homology of its nerve, thereby extending a result of Lafont and Métayer. Along the way, we investigate homotopy colimits with respect to the Folk model structure and deduce a theorem which formally resembles Quillen's Theorem A.

An extension of Quillen’s Theorem B

We prove a general version of Quillen's Theorem B, for actions of simplicial categories, in an arbitrary left Bousfield localization of the homotopy theory of simplicial presheaves over a site. As special cases, we recover a version of the group completion theorem in this general context, as well a version of Puppe's theorem on the stability of homotopy colimits in an infinity-topos, due to Rezk.

Cotorsion pairs and a K-theory localization theorem

We show that a complete hereditary cotorsion pair (C,C⊥) in an exact category E, together with a subcategory Z⊆E containing C⊥, determines a Waldhausen category structure on the exact category C, in which Z is the class of acyclic objects.This allows us to prove a new version of Quillen's Localization Theorem, relating the K-theory of exact categories A⊆B to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require A to be a Serre subcategory, which produces new examples.Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.

Stratification and duality for homotopical groups

We generalize Quillen's F-isomorphism theorem, Quillen's stratification theorem, the stable transfer, and the finite generation of cohomology rings from finite groups to homotopical groups. As a consequence, we show that the category of module spectra over C⁎(BG,Fp) is stratified and costratified for a large class of p-local compact groups G including compact Lie groups, connected p-compact groups, and p-local finite groups, thereby giving a support-theoretic classification of all localizing and colocalizing subcategories of this category. Moreover, we prove that p-compact groups admit a homotopical form of Gorenstein duality.

A Quillen Theorem B for strict ∞‐categories

We prove a generalization of Quillen's Theorem B to strict $\infty$-categories. More generally, we show that under similar hypothesis as for Theorem B, the comma construction for strict $\infty$-categories, that we introduced with Maltsiniotis in a previous paper, is the homotopy pullback with respect to Thomason equivalences. We give several applications of these results, including the construction of new models for certain Eilenberg-Mac Lane spaces.

On the Grothendieck construction for ∞-categories

We provide, among other things: (i) a Bousfield–Kan formula for colimits in ∞-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) ∞-categorical generalizations of Barwick–Kan's Theorem Bn and Dwyer–Kan–Smith's Theorem Cn (regarding homotopy pullbacks in the Thomason model structure, which themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of the simultaneous and interwoven functoriality of colimits (or dually, of limits) for natural transformations and for pullback along maps of diagram ∞-categories.

Unstable splittings in Hodge filtered Brown–Peterson cohomology

We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown-Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson's unstable splitting. As a consequence, we obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology for complex manifolds.

A quantitative version of Catlin-D’Angelo–Quillen theorem

Let $${f (z,\bar z)}$$ be a positive bi-homogeneous hermitian form on $${\mathbb{C}^n}$$ , of degree m. A theorem proved by Quillen and rediscovered by Catlin and D’Angelo states that for N large enough, $${\langle{z,\bar z}\rangle^Nf(z,\bar z)}$$ can be written as the sum of squares of homogeneous polynomials of degree m + N. We show this works for N ≥ C f ((n + m) log n)3 where C f has a natural expression in terms of coefficients of f, inversely proportional to the minimum of f on the sphere. The proof uses a semiclassical point of view on which 1/N plays a role of the small parameter h.

Orbit spaces, Quillen's Theorem A and Minami's formula for compact Lie groups

Orbit spaces, Quillen's Theorem A and Minami's formula for compact Lie groups

An Algebraic Proof of Quillen's Resolution Theorem for K1

AbstractIn his 1973 paper [4] Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of Nenashev's paper [3], we are able to give an algebraic proof of Quillen's Resolution Theorem for K1 of an exact category. We view this as an advance towards the goal of giving an essentially algebraic subject an algebraic foundation.

K(n)-Torsion-Free H-Spaces and P(n)-Cohomology

AbstractThe H-space that represents Brown–Peterson cohomology BPk(–) was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum P(n), by constructing idempotent operations in P(n)–cohomology P(n)k(–) in the style of Boardman–Johnson–Wilson; this relies heavily on the Ravenel–Wilson determination of the relevant Hopf ring. The resulting (i – 1)-connected H-spaces Yi have free connective Morava K-homology k(n)*(Yi), and may be built from the spaces in the Ω-spectrum for k(n) using only vn-torsion invariants.We also extend Quillen's theorem on complex cobordism to show that for any space X, the P(n)*-module P(n)*(X) is generated by elements of P(n)i(X) for i ≥ 0. This result is essential for the work of Ravenel–Wilson–Yagita, which in many cases allows one to compute BP–cohomology from Morava K-theory.

Le théorème de Quillen, d'adjonction des foncteurs dérivés, revisité

The aim of this Note is to present a very simple original, purely formal, proof of Quillen's adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an abstract adjunction theorem for ‘absolute’ derived functors. In contrast with all known proofs, the explicit construction of the derived functors is not used. The ‘absolute derived functors’ are a non-additive generalization of Deligne's ‘everywhere defined derived functors’. To cite this article: G. Maltsiniotis, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

A few localisation theorems

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 Quillen's theorem A. We give some applications in algebaic and birational geometry.

A note on a graded ring analogue of Quillen's theorem

Let S = ⊕ ∞ i=0 S i be a graded ring and let M be a finitely presentable S-module, which is locally extended from ( S 0) m , for all maximal ideals m of S 0. Then M is globally extended from S 0.

On the Geometry of 2-Categories and their Classifying Spaces

In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen's Theorem A.