Faithful Functor Research Articles

The weighted spectrum of a regular ring

We introduce a weighted spectrum 𝕊pec R for a commutative von Neumann regular ring R, consisting of the ordinary spectrum together with the stalks of the Pierce sheaf. Then 𝕊pec R is a weighted Stone space, and 𝕊pec is a faithful functor which reflects isomorphisms. We give a criterion for a weighted Stone space to be of the form 𝕊pec R. Conversely, we exhibit a class of regular rings for which 𝕊pec is fully faithful, and which thus are determined by their weighted spectrum.

Q-groupoids and their cohomology

We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a model for the Q-cohomology of the classifying space. As examples, we obtain models for equivariant Q- and orbifold Q-cohomology, and for equivariant Lie algebroid and orbifold Lie algebroid cohomology. We obtain double complexes associated to Poisson groupoids and groupoid-algebroid "matched pairs".

The category of MV-pairs

An MV-pair is a pair (B,G), where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain condition. Recently it was proved by one of the authors that for an MV-pair (B,G), ∼G is an effect-algebraic congruence and B/∼G is an MV-algebra. Moreover, every MV-algebra M can be represented by an MV-pair in this way. In this paper we show that one can define a suitable category of MV-pairs in such a way that there exist a faithful functor from the category of MV-algebras to the aforementioned category and a functor in the reversed direction.

The Geometry of Unitary 2-Representations of Finite Groups and their 2-Characters

Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are ‘categorified’ in this context: just as representations of groups correspond to equivariant line bundles, 2-representations of groups correspond to equivariant gerbes. We also show how the 2-character of a 2-representation can be made functorial with respect to morphisms of 2-representations. Under the geometric correspondence, the 2-character of a 2-representation corresponds to the geometric character of its associated equivariant gerbe. This enables us to show that the complexified 2-character is a unitarily fully faithful functor from the complexified Grothendieck category of unitary 2-representations to the category of unitary conjugation equivariant vector bundles over the group.

Analytic vectors in continuousp-adic representations

AbstractGiven a compactp-adic Lie groupGover a finite unramified extensionL/ℚplet GL/ℚpbe the product over all Galois conjugates ofG. We construct an exact and faithful functor from admissibleG-Banach space representations to admissible locallyL-analyticGL/ℚp-representations that coincides with passage to analytic vectors in the caseL=ℚp. On the other hand, we study the functor ‘passage to analytic vectors’ and its derived functors over general basefields. As an application we compute the higher analytic vectors in certain locally analytic induced representations.

$\mathcal{D}$-modules arithmétiques associés aux isocristaux surconvergents. Cas lisse

Let 𝒱 be a mixed characteristic complete discrete valuation ring, 𝒫 a separated smooth formal scheme over 𝒱, P its special fiber, X a smooth closed subscheme of P, T a divisor in P such that T X =T∩X is a divisor in X and 𝒟 𝒫 † ( † T) ℚ the tensorized with ℚ weak completion of the sheaf of differential operators on 𝒫 with overconvergent singularities along T. We construct a fully faithful functor denoted by sp X↪𝒫,T,+ from the category of isocrystal on X∖T X overconvergent along T X into the category of coherent 𝒟 𝒫 † ( † T) ℚ -modules with support in X. Next, we prove the commutation of sp X↪𝒫,T,+ with (extraordinary) inverse images and dual functors. These properties are compatible with Frobenius.

Equality of proofs for linear equality

This paper is about equality of proofs in which a binary predicate formalizing properties of equality occurs, besides conjunction and the constant true proposition. The properties of equality in question are those of a preordering relation, those of an equivalence relation, and other properties appropriate for an equality relation in linear logic. The guiding idea is that equality of proofs is induced by coherence, understood as the existence of a faithful functor from a syntactical category into a category whose arrows correspond to diagrams. Edges in these diagrams join occurrences of variables that must remain the same in every generalization of the proof. It is found that assumptions about equality of proofs for equality are parallel to standard assumptions about equality of arrows in categories. They reproduce standard categorial assumptions on a different level. It is also found that assumptions for a preordering relation involve an adjoint situation.

Resolution of the uniform lower bound problem in constructive analysis

AbstractIn a previous paper we constructed a full and faithful functor ℳ︁ from the category of locally compact metric spaces to the category of formal topologies (representations of locales). Here we show that for a real‐valued continuous function f, ℳ︁(f) factors through the localic positive reals if, and only if, f has a uniform positive lower bound on each ball in the locally compact space. We work within the framework of Bishop constructive mathematics, where the latter notion is strictly stronger than point‐wise positivity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Generic representations of orthogonal groups: The functor category [formula omitted

In this paper, we define the functor category F quad associated to F 2 -vector spaces equipped with a quadratic form. We show the existence of a fully faithful, exact functor ι : F → F quad , which preserves simple objects, where F is the category of functors from the category of finite-dimensional F 2 -vector spaces to the category of all F 2 -vector spaces. We define the subcategory F iso of F quad , which is equivalent to the product of the categories of modules over the orthogonal groups; the inclusion is a fully faithful functor κ : F iso → F quad which preserves simple objects.

A constructive and functorial embedding of locally compact metric spaces into locales

The paper establishes, within constructive mathematics, a full and faithful functor M from the category of locally compact metric spaces and continuous functions into the category of formal topologies (or equivalently locales). The functor preserves finite products, and moreover satisfies f ⩽ g if, and only if, M ( f ) ⩽ M ( g ) for continuous f , g : X → R . This makes it possible to transfer results between Bishop's constructive theory of metric spaces and constructive locale theory.

Twisted Fourier–Mukai functors

Due to a theorem by Orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of Fourier–Mukai type. We extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor. As an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties.

Semilinear representations of PGL

Let L be the function field of a projective space $$\mathbb{P}^{n}_{k} $$ over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf $$\mathcal{V}$$ on $$\mathbb{P}^{n}_{k} $$ is a collection of isomorphisms $$\mathcal{V} \to g^{ * } \mathcal{V}$$ for each g ∈ H satisfying the chain rule. We construct, for any n > 1, a fully faithful functor from the category of finite-dimensional L-semilinear representations of H extendable to the semigroup End(L/k) to the category of coherent H-sheaves on $$\mathbb{P}^{n}_{k} .$$ The paper is motivated by a study of admissible representations of the automorphism group G of an algebraically closed extension of k of countable transcendence degree undertaken in [4]. The semigroup End(L/k) is considered as a subquotient of G, hence the condition on extendability. In the appendix it is shown that, if $${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}}$$ is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinear $$ \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}{\text{ - representation}}$$ of degree one is an integral L-tensor power of $$\hbox{det}_{L} \Omega ^{1}_{{L/k}} .$$ It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.

An explicit construction for the Happel functor

An easy explicit construction is given for a full and faithful functor from the bounded derived category of modules over an associative algebra A to the stable category of the repetitive algebra of A. This construction simplifies the one given by Happel.

Naturally Full Functors in Nature

We introduce and discuss the notion of a naturally full functor. The definition is similar to the definition of a separable functor; a naturally full functor is a functorial version of a full functor, while a separable functor is a functorial version of a faithful functor. We study the general properties of naturally full functors. We also discuss when functors between module categories and between categories of comodules over a coring are naturally full.

FAITHFUL FUNCTORS FROM CANCELLATIVE CATEGORIES TO CANCELLATIVE MONOIDS WITH AN APPLICATION TO ABUNDANT SEMIGROUPS

We prove that any small cancellative category admits a faithful functor to a cancellative monoid. We use our result to show that any primitive ample semigroup is a full subsemigroup of a Rees matrix semigroup [Formula: see text] where M is a cancellative monoid and P is the identity matrix. On the other hand a consequence of a recent result of Steinberg is that it is undecidable whether a finite ample semigroup embeds as a full subsemigroup of an inverse semigroup.

A Full and Faithful Nerve for 2-Categories

The notion of geometric nerve of a 2-category (Street, J. Pure Appl. Algebra 49 (1987), 283–335) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax 2-functors give rise to homotopies between the corresponding simplicial maps. These facts allow us to prove a representation theorem of the general non-abelian cohomology of groupoids (classifying non-abelian extensions of groupoids) by means of homotopy classes of simplicial maps.

CATEGORY OF (POM)L-FUZZY GRAPHS AND HYPERGRAPHS

In this note by considering a complete lattice L, we define the notion of an L-Fuzzy hyperrelation on a given non-empty set X. Then we define the concepts of (POM)L-Fuzzy graph, hypergraph and subhypergroup and obtain some related results. In particular we construct the categories of the above mentioned notions, and give a (full and faithful) functor form the category of (POM)L-Fuzzy subhypergroups ((POM)L-Fuzzy graphs) into the category of (POM)L-Fuzzy hypergraphs. Also we show that for each finite objects in the category of (POM)L-Fuzzy graphs, the coproduct exists, and under a suitable condition the product also exists.

Concrete Data Structures as Games

A result by Curien establishes that filiform concrete data structures can be viewed as games. We extend the idea to cover all stable concrete data structures. This necessitates a theory of games with an equivalence relation on positions. We present a faithful functor from the category of concrete data structures to this new category of games, allowing a game-like reading of the former. It is possible to restrict to a cartesian closed subcategory of these games, where the function space does not decompose and the product is given by the usual tensor product construction. There is a close connection between these games and graph games.

Almost ff-universal and q-universal varieties of modular 0-lattices

\def\Bbb#1{{\mathbb#1}}A variety $\Bbb V$ of algebras of a finite type is almost $f\!f$-universal if there is a finiteness-preserving faithful functor $F:\Bbb G\rightarrow \Bbb V$ from the category $\Bbb G$ of all graphs and their compatible maps such

Homogeneous coordinates for algebraic varieties

We associate to every divisorial (e.g., smooth) variety X with only constant invertible global functions and finitely generated Picard group a Pic(X)-graded homogeneous coordinate ring. This generalizes the usual homogeneous coordinate ring of the projective space and constructions of Cox and Kajiwara for smooth and divisorial toric varieties. We show that the homogeneous coordinate ring defines in fact a fully faithful functor. For normal complex varieties X with only constant global functions, we even obtain an equivalence of categories. Finally, the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated Picard group turns out to be a Krull ring admitting unique factorization.