David Jaz Myers, David I. Spivak
Polynomial functors are sums of covariant representable functors from the category of sets to itself. They have a robust theory with many applications -- from operads and opetopes to combinatorial species. In this paper, we define a contravariant analogue of polynomial functors: Dirichlet functors. We develop the basic theory of Dirichlet functors, and relate them to their covariant analogues.
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We prove in ZFC that if $G$ is a (right) $R$-module such that the groups $\Hom_R(\prod_{i\in I}G_i,G)$ and $\prod_{i\in I}\Hom_R(G_i,G)$ are naturally isomorphic for all families of $R$-modules $(G_i)_{i\in I}$ then G=0. The result is valid even we restrict to families such that $G_i\cong G$ for all $i\in I$.
Given a locally presentable category together with a suitable functorial cylinder object, we construct model structures which are sensitive to the `direction' of the cylinder. We show that the Covariant and Contravariant model structures on simplicial sets as well as the coCartesian and Cartesian model structures on marked simplicial sets are examples of our formalism. In this setting, notions of final and initial maps and smooth and proper maps arise very naturally and we will identify these maps in the examples.
This dissertation is devoted to a study of adjunctions concerning categories enriched over a quantaloid Q (or Q-categories for short), with the following types of adjunctions involved: (1) adjoint functors between Q-categories; (2) adjoint distributors between Q-categories; (3) adjoint functors between categories consisting of Q-categories. For a small quantaloid Q and a distributor between Q-categories, two adjunctions between the Q-categories of contravariant and covariant presheaves are presented. These adjunctions respectively extend the fundamental construction of Isbell adjunctions and K…
Yoneda'e Lemma is about the canonical isomorphism of all the natural transformations from a given representable covariant (contravariant, reps.) functor (from a locally small category to the category of sets) to a covariant (contravariant, reps.) functor. In this note we point out that given any representable functor and any functor we have the canonical natural transformation from the given representable functor to the "subset" functor of the given functor, "collecting all the natural transformations".
We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring $R$ as the kernels of certain functors $(R\textbf{-Mod})^{\text{op}}\to\textbf{Ab}$ rather than of functors $R\textbf{-Mod}\to\textbf{Ab}$ which are given by a pp pair. This paper will give various algebraic characterisations of these functors in the case that $R$ is an artin algebra. Suppose that $R$ is an artin algebra. An additive functor $G:(R\textbf{-Mod})^{\text{op}}\to\textbf{Ab}$ preserves inverse limits and $G|_{(R\textbf{-mod})^{\text{op}}}:(R\textbf{-mod})^{\text{op}}\to\…
Artin glueings of toposes as adjoint split extensions
本文将frames的Artin glueings与伴随分裂扩张的对应关系扩展至toposes的2-范畴，证明其构成2-自然反变等价，并引入态射构造了范畴Ext(H,N)。
One can think of power series or polynomials in one variable, such as $P(x)=2x^3+x+5$, as functors from the category $\mathsf{Set}$ of sets to itself; these are known as polynomial functors. Denote by $\mathsf{Poly}_{\mathsf{Set}}$ the category of polynomial functors on $\mathsf{Set}$ and natural transformations between them. The constants $0,1$ and operations $+,\times$ that occur in $P(x)$ are actually the initial and terminal objects and the coproduct and product in $\mathsf{Poly}_{\mathsf{Set}}$. Just as the polynomial functors on $\mathsf{Set}$ are the copresheaves that can be written as …
We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.
For an arbitrary group $G$, a (semi-)Mackey functor is a pair of covariant and contravariant functors from the category of $G$-sets, and is regarded as a $G$-bivariant analog of a commutative (semi-)group. In this view, a $G$-bivariant analog of a (semi-)ring should be a (semi-)Tambara functor. A Tambara functor is firstly defined by Tambara, which he called a TNR-functor, when $G$ is finite. As shown by Brun, a Tambara functor plays a natural role in the Witt-Burnside construction. It will be a natural question if there exist sufficiently many examples of Tambara functors, compared to the wid…
Brown functors of directed graphs
该研究针对有向图的Brown函子展开，证明满足三类公理的有限有向图同伦范畴到阿贝尔群范畴的反变Brown函子均可表，且第一道路上同调函子属于有向图Brown函子，为有向图同调理论提供了新的理论支撑。
The reflexive completion of a category consists of the Set-valued functors on it that are canonically isomorphic to their double conjugate. After reviewing both this construction and Isbell conjugacy itself, we give new examples and revisit Isbell's main results from 1960 in a modern categorical context. We establish the sense in which reflexive completion is functorial, and find conditions under which two categories have equivalent reflexive completions. We describe the relationship between the reflexive and Cauchy completions, determine exactly which limits and colimits exist in an arbitrary…
We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi--locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi--locally presentable and we obtain a new proof of Brown representability theorem. Examples of functors which are not representable are also given.
V-graded categories and V-W-bigraded categories: Functor categories and bifunctors over non-symmetric bases
本文研究了在非对称基础上的V-分级范畴和双分级范畴，提出了一种适用于任意单向范畴V的函子范畴和双函子理论，探讨了V-分级模和预射影范畴的例子，并与Garner和López Franco的理论进行了比较。
Brown representability for directed graphs
该研究证明满足可加性与Mayer-Vietoris公理的、从有限有向图同伦范畴到阿贝尔群的反变函子具有可表示性。
Let C be a triangulated category with a Serre functor S and X a non-zero contravariantly finite rigid subcategory of C. Then X is cluster tilting if and only if the quotient category C/X is abelian and S(X)=X[2]. As an application, this result generalizes work by Beligiannis.
There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is naturally isomorphic to the identity, such that F restricts to the Zariski spectrum functor Spec on the category of commutative rings (in a compatible way with the natural isomorphism). Both of these no-go re…
Lallement functor is a weak right multiadjoint
本文定义了Lallement函子，并证明其为弱右多伴随，同时探讨了其与Płonka函子的关系。
In this paper we propose and lay the foundations of a functorial framework for representing signals. By incorporating additional category-theoretic relative and generative perspective alongside the classic set-theoretic measure theory the fundamental concepts of redundancy, compression are formulated in a novel authentic arrow-theoretic way. The existing classic framework representing a signal as a vector of appropriate linear space is shown as a special case of the proposed framework. Next in the context of signal-spaces as a categories we study the various covariant and contravariant forms o…
We introduce the new concept of cartesian module over a pseudofunctor $R$ from a small category to the category of small preadditive categories. Already the case when $R$ is a (strict) functor taking values in the category of commutative rings is sufficient to cover the classical construction of quasi-coherent sheaves of modules over a scheme. On the other hand, our general setting allows for a good theory of contravariant additive locally flat functors, providing a geometrically meaningful extension of Crawley-Boevey's Representation Theorem. As an application, we relate and extend some previ…
Dependent Optics
本文提出依赖光学统一框架，形式化多种双向数据访问器，定义依赖光学范畴并确立余积条件，引入Tambara表示对逆变函子分类，将profunctor编码推广至依赖情形。
Closing the category of finitely presented functors under images made constructive
本文构造了范畴 $\mathcal{Q}(\mathbf{P})$，形式化表示加法范畴中态射像的商，证明其等价于有限展示函子范畴在取像下封闭的子范畴，并将 Macaulay2 的数据结构推广到任意加法范畴以建模分次模与函子等对象。
We consider all Bott-Samelson varieties ${\rm BS}(s)$ for a fixed connected semisimple complex algebraic group with maximal torus $T$ as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms ${\rm BS}(s)\hookrightarrow{\rm BS}(s')$, where $s$ is a subsequence of $s'$. Every morphism of the new category induces a map between the $T$-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded $H^\bulle…
In this paper we construct an analogue of Lurie's "unstraightening" construction that we refer to as the "comprehension construction". Its input is a cocartesian fibration $p \colon E \to B$ between $\infty$-categories together with a third $\infty$-category $A$. The comprehension construction then defines a map from the quasi-category of functors from $A$ to $B$ to the large quasi-category of cocartesian fibrations over $A$ that acts on $f \colon A \to B$ by forming the pullback of $p$ along $f$. To illustrate the versatility of this construction, we define the covariant and contravariant Yon…
Hilbert series of representations of categories of $G$-sets
本文研究有限群 $G$ 的有限自由 $G$-集合范畴上反变函子的希尔伯特级数，证明其具有有理形式，且分母线性多项式的系数位于由 $G$ 的特征标表生成的域中。
Informally, ${\mathbb Z}_2^n$-manifolds are 'manifolds' with ${\mathbb Z}_2^n$-graded coordinates and a sign rule determined by the standard scalar product of their ${\mathbb Z}_2^n$-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ${\mathbb Z}_2^n$-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ${\mathbb Z}_2^n$-points, i.e., trivial ${\mathbb Z}_2^n$-manifolds for whic…
Polynomial Functors and Shannon Entropy
本文证明可将提取 Dirichlet 多项式香农熵的 rig 同态提升为基于笛卡尔多项式范畴的 rig 函子，通过三步映射过程并引入新的分配幺半结构，将熵几何化解释为矩形的对数长宽比。
We define the graph minor category and prove that the category of contravariant representations of the graph minor category over a Noetherian ring is locally Noetherian. This can be regarded as a categorification of the Robertson--Seymour graph minor theorem. In addition, we generalize Sam and Snowden's Gröbner theory of categories to the setting of pairs consisting of a category along with a functor to sets, and we apply this theory to the edge functor on the graph minor category. As an application, we study homology groups of unordered configuration spaces of graphs, improving upon various f…
Convergence of martingales via enriched dagger categories
本文通过拓扑丰富类别理论，首次证明了鞅和逆向鞅在均值下的收敛性，并展示了其在概率分析中的应用。
What kind of linearly distributive category do polynomial functors form?
本文研究多项式函子范畴 Poly 在 Dirichlet 积与替换积下形成的线性分配范畴结构，证明其为双闭 isomix LDC，并刻画其对偶对象、核及线性（余）幺半群等结构。
We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel o…
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