Class Of Morphisms Research Articles

Minimal fibrations and the organizing theorem of simplicial homotopy theory

Quillen showed that simplicial sets form a model category (with appropriate choices of three classes of morphisms), which organized the homotopy theory of simplicial sets. His proof is very difficult and uses even the classification theory of principal bundles. Thus, Goerss–Jardine appealed to topological methods for the verification. In this paper we give a new proof of this organizing theorem of simplicial homotopy theory which is elementary in the sense that it does not use the classifying theory of principal bundles or appeal to topological methods.

Purity of branch and critical locus

To a dominant morphism X/S→Y/S of Nœtherian integral S-schemes one has the inclusion CX/Y⊂BX/Y of the critical locus in the branch locus of X/Y. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials ΩX/Y, ΩX/S, and ΩY/S that imply bounds on the codimensions of CX/Y and BX/Y. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski–Nagata–Auslander, and Faltings and Grothendieck, and van der Waerdenʼs purity for birational morphisms.

Non-abelian extensions of Lie 2-algebras

In this paper, we introduce the notion of derivations of Lie 2-algebras and construct the associated derivation Lie 3-algebra. We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.

Simplicial descent categories

Let D be a category and E a class of morphisms in D . In this paper we study the question of how to transfer homotopic structure from the category of simplicial objects in D , Δ ∘ D , to D through a ‘good’ functor s : Δ ∘ D → D , which we call simple functor. For instance, the Bousfield–Kan homotopy colimit in a Quillen simplicial model category is a good simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natural structure of Brown category of cofibrant objects on Δ ∘ D . We use these facts to produce cofiber sequences on the localized category of D by E , which give rise to a natural Verdier triangulated structure in the stable case.

Concordant and Monotone Morphisms

Concordant-dissonant and monotone-light factorisation systems on categories, ways to construct them, and conditions for them to coincide, as well as their examples are studied in this article. These factorisation systems are constructed from a reflection induced from a ground adjunction and a specified prefactorisation system. Furthermore, we give additional conditions, under which the monotone-light and the concordant-dissonant factorisations coincide for sub-reflections of the induced reflection. The adjunctions given by right Kan extensions, from the category of presheaves on sets, turn out to be very well-behaved examples, provided they satisfy the cogenerating set condition, which allows to describe the four classes of morphisms in the reflective and concordant-dissonant (= monotone-light) factorisations. It is also noticed that the faithfulness of the composite of the left-adjoint with the Yoneda embedding can be seen as a generalisation of the cogenerating set condition. Using this generalisation it is possible to present a convenient simplified version of the sufficient conditions above for the case of an adjunction from the category of presheaves on sets into a cocomplete category, satisfying the faithfulness of the abovementioned composite. Then, the same is done for induced sub-reflections from categories of models of (limit) sketches; in particular this explains why the monotone-light factorisation for categories via preordered sets is just the restriction of the same factorisation for simplicial sets via ordered simplicial complexes.

Some categories associated with bases of the kalman algebra

We investigate relationships in the Kalman algebra, viewed as an algebra over the algebra induced by the coefficients of the characteristic polynomial of the state matrix. To this end we introduce the categories $ \mathcal{B}\mathcal{S}\mathcal{S} $ and $ \mathcal{B}\mathcal{S}{\mathcal{S}^{\text{strict}}} $ associated with relationships in some basis of the Kalman algebra and also with reconstruction of relationships from known fragments. On these categories we construct the structures of the symmetrical monoidal category induced by addition and multiplication in the Kalman algebra. We investigate the properties of some of the most important classes of morphisms, in particular, we describe the structure and the action of the automorphism group.

Order-preserving reflectors and injectivity

We investigate a Galois connection in poset enriched categories between subcategories and classes of morphisms, given by means of the concept of right-Kan injectivity, and, specially, we study its relationship with a certain kind of subcategories, the KZ-reflective subcategories. A number of well-known properties concerning orthogonality and full reflectivity can be seen as a particular case of the ones of right-Kan injectivity and KZ-reflectivity. On the other hand, many examples of injectivity in poset enriched categories encountered in the literature are closely related to the above connection. We give several examples and show that some known subcategories of the category of T 0 -topological spaces are right-Kan injective hulls of a finite subcategory.

Algebraic groups over the field with one element

In this paper we provide a first realization of an idea of Jacques Tits from a 1956 paper, which first mentioned that there should be a field of charactéristique une, which is now called \({\mathbb{F}_1}\), the field with one element. This idea was that every split reductive group scheme over \({\mathbb{Z}}\) should descend to \({\mathbb{F}_1}\), and its group of \({\mathbb{F}_1}\)-rational points should be its Weyl group. We connect the notion of a torified scheme to the notion of \({\mathbb{F}_1}\)-schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive group schemes as \({\mathbb{F}_1}\)-schemes. We endow the class of \({\mathbb{F}_1}\)-schemes with two classes of morphisms, one leading to a satisfying notion of \({\mathbb{F}_1}\)-rational points, the other leading to the notion of an algebraic group over \({\mathbb{F}_1}\) such that every split reductive group is defined as an algebraic group over \({\mathbb{F}_1}\). Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of GL(n) and Grassmann varieties are realized in this theory.

Infinite words with finite defect

In this paper, we provide a new characterization of uniformly recurrent words with finite defect based on a relation between the palindromic and factor complexity. Furthermore, we introduce a class of morphisms P ret closed under composition and we show that a uniformly recurrent word with finite defect is an image of a rich (also called full) word under a morphism of class P ret . This class is closely related to the well-known class P defined by Hof, Knill, and Simon; every morphism from P ret is conjugate to a morphism of class P.

Characteristic classes of Lie algebroid morphisms

We extend R. Fernandes' construction of secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism.

Bicompletion of metrically generated constructs

We develop a general technique to study completeness in a metrically generated construct X, the objects of which can be isomorphically described as sets endowed with a gauge of quasi metrics, saturated in a certain way. Our approach depends on a given functor F on X, describing the transition to some known topological construct A in which there already exists a completion theory described by some reflector R and for which also the associated firm class of morphisms \( \mathfrak{L} \)(R) is known. This functor will permit to “lift” the notion of completeness from A to X. We describe sufficient conditions on R and F, ensuring that the lifting produces a completion theory RF for the construct X0 of T 0 objects and that the associated firm class of morphisms can be derived from \( \mathfrak{L} \)(R). Application of this technique to concrete examples results in many interesting completion theories and provides us with a far better understanding of several existing ones.

Duality theorem for motives

A general duality theorem for the category of motives is established, with a short, simple, and self-contained proof. Introduction Recently, due to the active study of cohomological invariants in algebraic geometry, “transplantation” of classical topological constructions to the algebraic-geometrical “soil” seems to be rather important. In particular, it is very interesting to study topological properties of the category of motives. The concept of a motive was introduced by Alexander Grothendieck in 1964 in order to formalize the notion of universal (co-)homology theory (see the detailed exposition of Grothendieck’s ideas in [5]). For us, the principal example of this type is the category of motives DM−, constructed by Voevodsky [12] for algebraic varieties. The Poincare duality is a classical and fundamental result in algebraic topology that initially appeared in Poincare’s first topological memoir “Analysis Situs” [9] (as a part of the Betti numbers symmetry theorem proof). The proof of the general duality theorem for extraordinary cohomology theories apparently belongs to Adams [1]. Our purpose in this paper is to establish a general duality theorem for the category of motives. Essentially, we extend the main statement of [8] to this category. Many known results can easily be interpreted in these terms. In particular, we get a generalization of the Friedlander–Voevodsky duality theorem [4] to the case of the ground field of arbitrary characteristic. The proof of this fact, involving the main result of [8], was kindly conveyed to the authors by Andrěi Suslin in a private communication. Being inspired by his work and Dold–Puppe’s category approach [2] to the duality phenomenon in topology, we decided to present a short, simple, and self-contained proof of a similar result for the category of motives. Our result might be viewed as a purely abstract theorem and rewritten in the spirit of “abstract nonsense” as a statement about some category with a distinguished class of morphisms. Essentially, what is required for the proof is the existence of finite fiber products and the terminal object in the category of varieties, a small part of the tensor triangulated category structure for motives, and finally, the existence of transfers for the class of morphisms generated by graphs of a special type (of projective morphisms). However, rather, we preferred to formulate all statements for motives of algebraic varieties in order to clarify the geometric nature of the construction and make possible applications easier. This led, in particular, to the appearence of the second (co)homology index responsible for twist with the Tate object Z(1) (see Voevodsky [12]). The only exception is the classical Example 2. 2000 Mathematics Subject Classification. Primary 14F42.

Noneffective Regularity of Equality Languages and Bounded Delay Morphisms

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications We give an instance of a class of morphisms for which it is easy to prove that their equality set is regular, but its emptiness is still undecidable. The class is that of bounded delay 2 morphisms.

Connected and Disconnected Maps

A new relation between morphisms in a category is introduced—roughly speaking (accurately in the categories Set and Top), f ∥ g iff morphisms w:dom(f)→dom(g) never map subobjects of fibres of f non-constantly to fibres of g. (In the algebraic setting replace fibre with kernel.) This relation and a slight weakening of it are used to define “connectedness” versus “disconnectedness” for morphisms. This parallels and generalises the classical treatment of connectedness versus disconnectedness for objects in a category (in terms of constant morphisms). The central items of study are pairs \(({\mathcal F},{\mathcal G})\) of classes of morphisms which are corresponding fixed points of the polarity induced by the ∥-relation. Properties of such pairs are examined and in particular their relation to (pre)factorisation systems is analysed. The main theorems characterise: (a) factorisation systems which factor morphisms through a regular epimorphic “connected” morphism followed by a “disconnected” morphism, and (b) pairs \(({\mathcal F},{\mathcal G})\) consisting of “connected” versus “disconnected” morphisms which induce a (regular) factorisation system. This suggests a generalisation of the pair (Concordant, Dissonant) of classes of continuous maps which was shown by Collins to yield the factorisation system (Concordant quotient, Dissonant) on Top.

UNIVERSAL SPACES, TYCHONOFF AND SPECTRAL SPACES

This paper deals with some universal spaces. The class of morphisms in TOP orthogonal to all Tychonoff spaces is characterized. We also characterize topological spaces X for which the universal Tychonoff space associated to X is a spectral space.

Categorical Abstract Algebraic Logic: Structurality, protoalgebraicity, and correspondence

AbstractThe notion of an ℐ ‐matrix as a model of a given π ‐institution ℐ is introduced. The main difference from the approach followed so far in Categorical Abstract Algebraic Logic (CAAL) and the one adopted here is that an ℐ ‐matrix is considered modulo the entire class of morphisms from the underlying N ‐algebraic system of ℐ into its own underlying algebraic system, rather than modulo a single fixed (N,N ′)‐logical morphism. The motivation for introducing ℐ ‐matrices comes from a desire to formulate a correspondence property for N ‐protoalgebraic π ‐institutions closer in spirit to the one for sentential logics than that considered in CAAL before. As a result, in the previously established hierarchy of syntactically protoalgebraic π ‐institutions, i. e., those with an implication system, and of protoalgebraic π ‐institutions, i. e., those with a monotone Leibniz operator, the present paper interjects the class of those π ‐institutions with the correspondence property, as applied to ℐ ‐matrices. Moreover, this work on ℐ ‐matrices enables us to prove many results pertaining to the local deduction‐detachment theorems, paralleling classical results in Abstract Algebraic Logic formulated, first, by Czelakowski and Blok and Pigozzi. Those results will appear in a sequel to this paper. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Orthogonal Subcategory Problem and the Small Object Argument

A classical result of P. Freyd and M. Kelly states that in “good” categories, the Orthogonal Subcategory Problem has a positive solution for all classes ${\mathcal {H}}$ of morphisms whose members are, except possibly for a subset, epimorphisms. We prove that under the same assumptions on the base category and on ${\mathcal {H}}$ , the generalization of the Small Object Argument of D. Quillen holds—that is, every object of the category has a cellular ${\mathcal {H}}$ -injective weak reflection. In locally presentable categories, we prove a sharper result: a class of morphisms is called quasi-presentable if for some cardinal λ every member of the class is either λ-presentable or an epimorphism. Both the Orthogonal Subcategory Problem and the Small Object Argument are valid for quasi-presentable classes. Surprisingly, in locally ranked categories (used previously to generalize Quillen’s result), this is no longer true: we present a class ${\mathcal {H}}$ of morphisms, all but one being epimorphisms, such that the orthogonality subcategory ${\mathcal {H}}^\perp$ is not reflective and the injectivity subcategory Inj $\,{\mathcal {H}}$ is not weakly reflective. We also prove that in locally presentable categories, the injectivity logic and the Orthogonality Logic are complete for all quasi-presentable classes.

Cover Relations on Categories

A cover relation on a category ℂ is a binary relation ⊏ on the class of morphisms of ℂ, which is defined only for those pairs of morphisms which have the same codomain, and which has the following two properties: (i) if f ⊏ g and h is composable with f, then hf ⊏ hg, (ii) if f ⊏ g and f is composable with e then fe ⊏ g. We study cover relations arising from a special type of factorization systems, and cover relations arising from a special type of monoidal structures.

Functors of Lindenbaum-Tarski, Schematic Interpretations, and Adjoint Cylinders between Sentential Logics

We prove, by using the concept of schematic interpretation, that the natural embedding from the category ISL, of intuitionistic sentential pretheories and i-congruence classes of morphisms, to the category CSL, of classical sentential pretheories and c-congruence classes of morphisms, has a left adjoint, which is related to the double negation interpretation of Gödel-Gentzen, and a right adjoint, which is related to the Law of Excluded Middle. Moreover, we prove that from the left to the right adjoint there is a pointwise epimorphic natural transformation and that since the two endofunctors at CSL, obtained by adequately composing the aforementioned functors, are naturally isomorphic to the identity functor for CSL, the string of adjunctions constitutes an adjoint cylinder. On the other hand, we show that the operators of Lindenbaum-Tarski of formation of algebras from pretheories can be extended to equivalences of categories from the category CSL, respectively, ISL, to the category Bool, of Boolean algebras, respectively, Heyt, of Heyting algebras. Finally, we prove that the functor of regularization from Heyt to Bool has, in addition to its well-known right adjoint (that is, the canonical embedding of Bool into Heyt) a left adjoint, that from the left to the right adjoint there is a pointwise epimorphic natural transformation, and, finally, that such a string of adjunctions constitutes an adjoint cylinder.

Uniqueness of completion for metrically generated constructs

In this paper, for metrically generated constructs X in the sense of [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547–1556] we study completion as a U -reflector R on the subconstruct X 0 of all T 0 -objects, for U some class of embeddings. Roughly speaking we deal with constructs X that are generated by the subclass of their metrizable objects and for various types of completion functors R available in that context, we obtain internal descriptions of the largest class U for which completion is unique. We apply our results to some well known situations. Completion of uniform spaces, of proximity spaces or of non-Archimedian uniform spaces is unique with respect to the class of all epimorphic embeddings, and this class is the largest one. However the largest class of morphisms for which Dieudonné completion of completely regular spaces or of zero dimensional spaces is unique, is strictly smaller than the class of all epimorphic embeddings. The same is true for completion in quantitative theories like uniform approach spaces for which the largest U coincides with the class of all embeddings that are dense with respect to the metric coreflection. Our results on completion for metrically generated constructs explain these differences.