Generic representations of orthogonal groups: projective functors in the category Fquad

An $\infty$-categorical localisation functor for diagrams of simplicial sets

Associated to each small category $C$, there is a category of $C$-shaped diagrams of simplicial sets and an $\infty$-category of $NC$-shaped homotopy coherent diagrams of spaces. We present a functor which exhibits the latter as the $\infty$-categorical localisation of the former at the objectwise weak homotopy equivalences. This builds on a Quillen equivalence between the projective and covariant model structures associated to $C$ due to Heuts-Moerdijk, as well as Cisinski's theory of $\infty$-categorical localisations. We use the localisation functor to give simplified proofs that the left (…

格路上的泛函的交叉效应

Cross effects for functors from posets

本文研究了格路上泛函的泛函微分学与多参数持久模块的投射维数关系，提出了新的泛函微分学概念，并证明了多参数持久模块投射维数的充要条件。

A Pirashvili-type theorem for functors on non-empty finite sets

Pirashvili's Dold-Kan type theorem for finite pointed sets follows from the identification in terms of surjections of the morphisms between the tensor powers of a functor playing the role of the augmentation ideal; these functors are projective. We give an unpointed analogue of this result: namely, we compute the morphisms between the tensor powers of the corresponding functor in the unpointed context. We also calculate the Ext groups between such objects, in particular showing that these functors are not projective; this is an important difference between the pointed and unpointed contexts. T…

Cubical Resolutions and Derived Functors

We introduce pseudocubical objects with pseudoconnections in an arbitrary category, obtained from the Brown-Higgins structure of a cubical object with connections by suitably relaxing their identities, and construct a cubical analog of the Tierney-Vogel theory of simplicial derived functors. The crucial point in the construction is that projective precubical resolutions which are naturally used to define our cubical derived functors possess pseudodegeneracies and pseudoconnections. The same fact is essentially used for proving that in the case of an additive functor between abelian categories,…

The block structure spaces of real projective spaces and orthogonal calculus of functors II

For a finite dimensional real vector space V with inner product, let F(V) be the block structure space, in the sense of surgery theory, of the projective space of V. Continuing a program launched in part I, we investigate F as a functor on vector spaces with inner product, relying on functor calculus ideas. It was shown in part I that F agrees with its first Taylor approximation T_1 F (which is a polynomial functor of degree 1) on vector spaces V with dim(V) > 5. To convert this theorem into a functorial homotopy-theoretic description of F(V), one needs to know in addition what T_1 F(V) is whe…

A pair of homotopy-theoretic version of TQFT's induced by a Brown functor

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT's starting from a $\mathsf{Hopf}^\mathsf{bc}_k$-valued Brown functor where $\mathsf{Hopf}^\mathsf{bc}_k$ is the category of bicommutative Hopf algebras over a field $k$ : the cospanical path-integral and the spanical path-…

Lannes' t functor on injective unstable modules and harish-chandra restriction

In the 1980's, the magic properties of the cohomology of elementary abelian groups as modules over the Steenrod algebra initiated a long lasting interaction between topology and modular representation theory in natural characteristic. The Adams-Gunawardena-Miller theorem in particular, showed that their decomposition is governed by the modular representations of the semi-groups of square matrices. Applying Lannes' T functor on the summands L P := Hom Mn(Fp) (P, H * (F p) n) defines an intriguing construction in representation theory. We show that T(L P) $\sim$ = L P $\oplus$ H * V 1 $\otimes$ …

A family of acyclic functors

We determine a family of functors from a poset to abelian groups such that the higher direct limits vanish on them. This is done by first characterizing the projective functors. Then a spectral sequence arising from the grading of the poset is used. Also the dual version for injective functors and higher inverse limits is included. Graded posets include simplicial complexes, subdivision categories and simplex-like posets.

Constructing cogenerators in triangulated categories and Brown representability

For a triangulated category with products we develop a method for constructing a nice set of cogenerators, allowing us to prove a formal criterion in order to satisfy Brown representability for covariant functors. We apply this criterion for showing that both homotopy category of projective modules and homotopy category of projective objects in a category of functors from a small category to a module category satisfy this kind of representability. In particular, homotopy category of projective complexes satisfies Brown representability for covariant functors.

Equivalences between localisations of categories provided by replacements

We give a characterisation of functors whose induced functor on the level of localisations is an equivalence and where the isomorphism inverse is induced by some kind of replacements such as projective resolutions or cofibrant replacements.

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the $n$-homogeneous model structure, the $n$-th derivative is a Quillen functor to the category of spectra with $Σ_n$-action. After taking into account only finitary functors -- which …

解构奥斯特兰德公式，I. 与加法函子相关的基本序列

Deconstructing Auslander's formulas, I. Fundamental sequences associated with additive functors

该研究针对加法函子构造长正合序列，半正合函子所得序列正合，一般函子非平凡同调可能仅在导出函子处。专门针对有限表现模上的Hom和张量积函子恢复奥斯特兰德公式，结果适用于任意环和模，还得出任意复形同调与上同调的通用系数定理。

On the Inadequacy of the Projective Structure with Respect to the Univalence Axiom

In this article the author endows the functor category [B(C2),Gpd] with the structure of a type-theoretic fibration category with a universe using the projective fibrations. It offers a new model of Martin-Löf type theory with dependent sums, dependent products, identity types and a universe. It turns out that this universe, the natural candidate that lifts the univalent universe of small discrete groupoids in the groupoid model of Hofmann and Streicher, is not univalent.

驯顺函子的阿贝尔结构与模型结构

Abelian and model structures on tame functors

本文研究了驯顺函子范畴继承阿贝尔结构与模型结构的条件，提出了余纤维对象结构定理，并引入了生成不可分解对象的通用技术。

有限集范畴上的函子再探讨

Functors on the category of finite sets revisited

本文在特征零域上研究了有限集与所有映射范畴 $\mathbf{FA}$ 的表示范畴结构，构造了单表示与不可分解投射对象，利用投射生成元证明了 Morita 等价，并提供了基于 $\mathbf{FB}$ 表示计算合成因子重数及态射空间的方法。