Category Of Functors Research Articles

Enriched ∞-categories via non-symmetric ∞-operads

We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal ∞-category. Our theory of enriched ∞-categories has many desirable properties; for instance, if the enriching ∞-category V is presentably symmetric monoidal then Cat∞V is as well. These features render the theory useful even when an ∞-category of enriched ∞-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and (∞,n)-categories). This is analogous to the advantages of ∞-categories over more rigid models such as simplicial categories — for example, the resulting ∞-categories of functors between enriched ∞-categories automatically have the correct homotopy type.We construct the homotopy theory of V-enriched ∞-categories as a certain full subcategory of the ∞-category of “many-object associative algebras” in V. The latter are defined using a non-symmetric version of Lurie's ∞-operads, and we develop the basics of this theory, closely following Lurie's treatment of symmetric ∞-operads. While we may regard these “many-object” algebras as enriched ∞-categories, we show that it is precisely the full subcategory of “complete” objects (in the sense of Rezk, i.e. those whose spaces of objects are equivalent to their spaces of equivalences) that are local with respect to the class of fully faithful and essentially surjective functors. We also consider an alternative model of enriched ∞-categories as certain presheaves of spaces satisfying analogues of the “Segal condition” for Rezk's Segal spaces. Lastly, we present some applications of our theory, most notably the identification of associative algebras in V as a coreflective subcategory of pointed V-enriched ∞-categories as well as a proof of a strong version of the Baez–Dolan stabilization hypothesis.

A categorical view of textural approximation spaces

The basic motivation of the theory of textures is to find a convenient point-set based setting for fuzzy sets. Recent works on textures show that they also provide a useful model for rough set theory. In this paper, we show that i-c spaces (interior-closure texture spaces) can be regarded as a textural rough set systems on a single universe. Then we consider the approaches containing direlations and dicovers for textural rough sets and we give some basic results related to direlations and dicovers. Considering the discrete textures we discuss on these results for rough sets based on relations and coverings. Finally, we prove that the category of topological spaces and continuous functions is isomorphic to the category of (covering) approximation spaces and continuous functions.

Points in algebraic geometry

We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, étale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type over a separated noetherian base. Combined with a theorem of Deligne on the existence of enough points, this provides an algebro-geometric description of a conservative family of fibre functors on these categories of sheaves. As an example of an application we show that direct image along a closed immersion is exact for all these topologies except qfh. The methods are transportable to other categories of sheaves as well.

Combinatorial categorical equivalences of Dold–Kan type

We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let X denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold–Puppe–Kan theorem that simplicial objects in X are equivalent to chain complexes in X; (b) the observation of Church, Ellenberg and Farb [9] that X-valued species are equivalent to X-valued functors from the category of finite sets and injective partial functions; (c) a result T. Pirashvili calls of “Dold–Kan type”; and so on. When X is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourne.

A local-global principle for small triangulated categories

AbstractLocal cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category ⊺ equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for ⊺ and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories.

Leibniz homology of Lie algebras as functor homology

We prove that Leibniz homology of Lie algebras can be described as functor homology in the category of linear functors from a category associated with the Lie operad.

Trace maps for Mackey algebras

Let G be a finite group and R be a commutative ring. The Mackey algebra μR(G) shares a lot of properties with the group algebra RG however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetry of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra.Using the fact that the category of Mackey functors is a closed symmetric monoidal category, we prove that the Mackey algebra μR(G) is a symmetric algebra if and only if the family of Burnside algebras (RB(H))H⩽G is a family of symmetric algebras with a compatibility condition.As a corollary, we recover the well known fact that over a field of characteristic zero, the Mackey algebra is always symmetric. Over the ring of integers the Mackey algebra of G is symmetric if and only if the order of G is square free. Finally, if (K,O,k) is a p-modular system for G, we show that the Mackey algebras μO(G) and μk(G) are symmetric if and only if the Sylow p-subgroups of G are of order 1 or p.

Generic representation theory of finite fields in nondescribing characteristic

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and charF is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all n≥0, of the categories of K[GLn(F)]-modules.As a consequence, if K is also a field, then small projectives are also injective in Rep(F;K), and will have finite length. Even more is true if charK=0: the category Rep(F;K) will be semisimple.In the last section, we briefly discuss ‘q=1’ analogues and consider representations of various categories of finite sets.The main result follows from a 1992 result by L.G. Kovács about the semigroup ring K[Mn(F)].

Linear filters and hereditary torsion theories in functor categories

We introduce the notion of Gabriel filter for a preadditive category C and we show that there is a bijective correspondence between Gabriel filters of C and hereditary torsion theories in the category of additive functors (C,Ab), obtaining a generelization of the theorem given by Gabriel [Ga] and Maranda [Ma] which establishes a bijective correspondence between Gabriel filters for a ring and hereditary torsion theories in the corresponding category of modules.

Deriving Auslander's formula

Auslander's formula shows that any abelian category \mathsf C is equivalent to the category of coherent functors on \mathsf C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of \mathsf C ) is compactly generated and that the full subcategory of compact objects is equivalent to the bounded derived category of \mathsf C . The same approach shows for an arbitrary Grothendieck abelian category that its derived category and the homotopy category of injective objects are well-generated triangulated categories. For sufficiently large cardinals \alpha we identify their \alpha -compact objects and compare them.

Derived Kan Extension for Strict Polynomial Functors

We investigate fundamental properties of adjoint functors to the precomposition functor in the category of strict polynomial functors.

A classification of Taylor towers of functors of spaces and spectra

We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which we construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this comonad.For functors with values in the category of spectra, we give a rather explicit description of the associated comonads and their coalgebras. In particular, for functors from based spaces to spectra we interpret this new structure as that of a divided power right module over the operad formed by the derivatives of the identity on based spaces.

The ring of polynomial functors of prime degree

LetZ^p\hat {\mathbb {Z}}_pbe the ring ofpp-adic integers. We prove in the present paper that the category of polynomial functors from finitely generated free abelian groups toZ^p\hat {\mathbb {Z}}_p-modules of degree at mostppis equivalent to the category of modules over a particularly well understood ring, called Green order. This case was conjectured by Yuri Drozd.

Category Theory is a Contentful Theory

In (Linnebo & Pettigrew, 2011), some objections to category theory as an autonomous foundation are presented. The authors of that paper do a commendable job making clear several distinct senses of “autonomous” as it occurs in the phrase “autonomous foundation.” Unfortunately, the paper seems to treat the “categorist” perspective rather unfairly. Several infelicities of this sort were addressed by (McLarty, 2012). I wish in this note to address yet another apparent infelicity. The subject of this paper is the comments in (Linnebo & Pettigrew, 2011) concerning the contentfulness of William Lawvere’s axiomatic system CCAF. For details of this system itself the reader is encouraged to consult (Lawvere, 1966) or (Lawvere, 1963). No technical details from these expositions will be needed, however. The authors of (Linnebo & Pettigrew, 2011) are willing to admit “CCAF asserts the existence of certain categories and describes some of the functors between them.” However, as they correctly point out, this by itself is insufficient for CCAF to serve as a foundation of mathematics. As the authors point out, a necessary condition for CCAF to serve as a foundation of mathematics is that be contentful, and, “if it is to make a contentful assertion, we need to be able to identify its subject matter – namely, categories – independently of the theory.” The authors provide us an illustrative example of a non-contentful theory to help make this objection more clear. “[A lack of content] would be the problem, for instance, with the theory that consists of the following sentence: ‘the mome raths outgrabe’. The reason that this theory lacks content is that there is no way of identifying, independently of the theory, what mome raths are, or what it is to outgrabe. Without identifying the subject matter of the theory in such a way, the statements of the theory do not make contentful assertions.” (Linnebo & Pettigrew, 2011, p. 231) Presumably (and the authors are certainly correct here) a theory that runs afoul this objection (which I will call the Contentful Theory Objection [CTO]) cannot serve as an autonomous foundation for mathematics. Of course, a proponent of CCAF can respond to CTO by simply providing a CCAF-free identification of the subject matter of category theory. The problem, the authors hold, is that this cannot be accomplished without running afoul what they call the Logical Dependence Objection [LDO]: if any theory T is to provide an alternative foundation for mathematics to the foundation provided by set theory, it must not be the case that T “depend[s] logically on a prior theory of classes and functions in order to ground [its] existential assertions.” That is, if one cannot identify This work was supported by the National Science Foundation under Grant No. 00006595. A side note: CCAF is often assumed to abbreviate either “category of categories and functors” or “category of categories as foundation.” In Lawvere’s thesis, however, the system normally called CCAF appears under the heading “category of categories and adjoint functors”.

Inducing native Mackey functors to biset functors

In this paper, we describe the induction functor from the category of native Mackey functors to the category of biset functors for a finite group G over an algebraically closed field k of characteristic zero. We prove two applications of this description. As the first application, we exhibit that any projective biset functor over k is induced from a (rational) virtual native Mackey functor. The second application is the explicit description of the projective indecomposable biset functors parameterized by simple groups.

Gelfand spectra in Grothendieck toposes using geometric mathematics

In the (covariant) topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to each unital C*-algebra, A, a topos T(A) of sheaves on a locale and a commutative C*-algebra, a, within that topos. The Gelfand spectrum of a is a locale S in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale S, by noting that the construction of the Gelfand spectrum in a general topos can be described using geometric logic. As a consequence, the spectrum, seen as a bundle, is computed fibrewise. As a by-product of the geometricity of Gelfand spectra, we find an explicit external description of the spectrum whenever the topos is a functor category. As an intermediate result we show that locally perfect maps compose, so that the externalization of a locally compact locale in a topos of sheaves over a locally compact locale is locally compact, too.

Fused Mackey functors

Let \(G\) be a finite group. In [6], the question of comparing Mackey functors for \(G\) and biset functors defined on subgroups of \(G\) and bifree bisets as morphisms has been considered. This paper proposes a different approach to this problem, from the point of view of various categories of \(G\)-sets. In particular, the category \(G\hbox {-}{\underline{\mathsf{set}}}\) of fused\(G\)-sets is introduced, as well as the category \(\underline{\mathbf {S}}(G)\) of spans in \(G\hbox {-}{\underline{\mathsf{set}}}\). The fused Mackey functors for \(G\) over a commutative ring \(R\) are defined as \(R\)-linear functors from \(R\,\underline{\mathbf {S}}(G)\) to \(R\)-modules. They form an abelian subcategory \(\mathsf {Mack}_R^f(G)\) of the category of Mackey functors for \(G\) over \(R\). The category \(\mathsf {Mack}_\mathbb {Z}^f(G)\) is equivalent to the category of conjugation Mackey functors of [6]. The category \(\mathsf {Mack}_R^f(G)\) is also equivalent to the category of modules over the fused Mackey algebra\(\mu _R^f(G)\), which is a quotient of the usual Mackey algebra \(\mu _R(G)\) of \(G\) over \(R\).

On Transferring Model Theoretic Theorems of $${\mathcal{L}_{{\infty},\omega}}$$ L ∞ , ω in the Category of Sets to a Fixed Grothendieck Topos

Working in a fixed Grothendieck topos Sh(C, JC) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, JC). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\), such as the downward Lowenheim–Skolem theorem, the completeness theorem and Barwise compactness.

A Linear Category of Polynomial Functors (extensional part)

We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is reminiscent of Day's convolution on presheaves. We then make this category into a model for intuitionistic linear logic by defining an additive and exponential structure.

Theories of analytic monads

In this paper we characterise the categories of Lawvere theories and equational theories that correspond to the categories of analytic and polynomial monads onSet, and hence also to the categories of the symmetric and rigid operads inSet. We show that the category of analytic monads is equivalent to the category of regular-linear theories. The category of polynomial monads is equivalent to the category of rigid theories, that is, regular-linear theories satisfying an additional global condition. This solves a problem posed by A. Carboni and P. T. Johnstone. The Lawvere theories corresponding to these monads are identifiedviasome factorisation systems. We also show that the categories of analytic monads and finitary endofunctors onSetare monadic over the category of analytic functors. The corresponding monad for analytic monads distributes over the monad for finitary endofunctors and hence the category of (finitary) monads onSetis monadic over the category of analytic functors. This extends a result of M. Barr.