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We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings. We apply our results to corings coming from entwining structures and graded structures, and we obtain new results in the setting of entwining structures and in the graded ring theory.

We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I^0 equivariant categories, where I^0 is the radical of the Iwahori subgroup of G((t)). Our result may be viewed as an affine analogue of the equivalence of categories of g-modules and D-modules on the flag variety G/B, due to Beilinson-Bernstein and Brylinski-Kashiwara., Comment: 33 pages

International audience; I will discuss the two problems of how to define identity between logics and how to define identity between proofs. For the identity of logics, I propose to simply use the notion of preorder equivalence. This might be considered to be folklore, but is exactly what is needed from the viewpoint of the problem of the identity of proofs: If the proofs are considered to be part of the logic, then preorder equivalence becomes equivalence of categories, whose arrows are the proofs. For identifying these, the concept of proof nets is discussed.

The (dual) Dold-Kan correspondence says that there is an equivalence of categories K: Ch≥0→AbΔ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR*→RingsΔ, although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR* and RingsΔ are Quillen closed model categories and the total left derived functor of K is an equivalence: LK: Ho DGR* Ho RingsΔ. The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through different methods. Our proof is based on a functor Q:DGR*→RingsΔ, naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-Kostant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module [n] ∐nRS that arises from a homomorphism R→S of not necessarily commutative rings, using the coproduct ∐R of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S⊗Rn originally defined by Nuss K-theory 12 (1997) 23, using braids., Facultad de Ciencias Exactas

For a pair of affine toric varieties X and Y defined by dual cones, we define an equivalence between two triangulated categories. The first is a mixed version of the equivariant derived category of X and the second is a mixed version of the derived category of sheaves on Y which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel. A similar duality was constructed in math.AG/0308216; this new approach is more canonical., Comment: 47 pages

Certain quantization problems are equivalent to the construction of morphisms from “quantum” to “classical” props. Once such a morphism is constructed, Hensel's lemma shows that it is in fact an isomorphism. This gives a new, simple proof that any Etingof–Kazhdan quantization functor is an equivalence of categories between quantized universal enveloping (QUE) algebras and Lie bialgebras over a formal series ring (dequantization). We apply the same argument to construct dequantizations of formal solutions of the quantum Yang–Baxter equation and of quasitriangular QUE algebras. We derive from there a classification of all twistors killing a given associator. We also give structure results for the props involved in quantization of Lie bialgebras, which yield an associator-independent proof that the prop of QUE algebras is a flat deformation of the prop of co-Poisson universal enveloping algebras.

The authors report the implementation and evaluation of a function-based recognition system that takes an uninterrupted 3-D object shape as its input and reasons to determine if the object belongs to the superordinate category furniture, and if so, into which (sub)category it falls. The system has analyzed over 250 input objects, and the results largely agree with intuitive human interpretation of the objects. The study confirms that a relatively small number of knowledge primitives may be used as the basis for defining a relatively broad range of object categories. The greatest derivation from intuitive human interpretation occurs with objects that humans would not typically label as one of the known categories defined, but which have some novel orientation in which they could serve the function of one of these categories. This is because the system uses a purely function-based definition of the object category. >

Let $(H, \a)$ be a monoidal Hom-Hopf algebra and $(A, \b)$ a right $(H, \a)$-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of $(A, \b)$ in the setting of monoidal Hom-Hopf algebras. Also we prove that there exists a total integral $\phi: (H, \a)\rightarrow (A, \b)$ if and only if any representation of the pair $(H,A)$ is injective in a functorial way, as a corepresentation of $(H, \a)$, which generalizes Doi's result. Finally, we define a total quantum integral $\g: H\rightarrow Hom(H, A)$ and prove the following affineness criterion: if there exists a total quantum integral $\g$ and the canonical map $\psi: A\o_{B}A\rightarrow A\o H,\ \ a\o_{B}b\mapsto \b^{-1}(a)b_{[0]}\o \a(b_{[1]}) $is surjective, then the induction functor $A\o_B-: \widetilde{\mathscr{H}}(\mathscr{M}_k)_{B}\rightarrow \widetilde{\mathscr{H}}(\mathscr{M}_k)^{H}_{A}$ is an equivalence of categories., Comment: 20 pages. arXiv admin note: text overlap with arXiv:math/0106067 by other authors

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories., Comment: 22 pages

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any modular category (not necessarily hermitian) is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category and classify categories of prime dimension., Comment: 50 pages, latex; a reference to Schauenburg's freeness theorem was added and a new section 5.11 to fill a gap in the previous version; also Example 7.2 was corrected; in April 2017 a comment added in Subsection 9.3 giving a reference which fills a gap in this subsection

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel and Hovey-Sadofsky change of rings theorems in algebraic topology., 25 pages

First let us recall some results about a decomposition of projective modules and an equivalence of categories which will be used in the sequel.

Sesquicartesian categories are categories with nonempty finite products and arbitrary finite sums, including the empty sum. Coherence is here demonstrated for sesquicartesian categories in which the first and the second projection from the product of the initial object with itself are the same. (Every bicartesian closed category, and, in particular, the category Set, is such a category.) This coherence amounts to the existence of a faithful functor from categories of this sort freely generated by sets of objects to the category of relations on finite ordinals. Coherence also holds for bicartesian categories where, in addition to this equality for projections, we have that the first and the second injection to the sum of the terminal object with itself are the same. These coherences yield a very easy decision procedure for equality of arrows.

Weak equivalence is defined as equivalence in the bicategory of modules between internal categories. It is known that two categories are weakly equivalent if and only if their Cauchy completions are equivalent. We prove that this condition can be generalized to a suitable notion of intermediate category, stable under composition with weak equivalences. Applications to categorical Morita theory are given.

To any additive category $\C$, we associate in a functorial way two additive categories $\mathcal A(\C)$, $\mathcal B(\C)$. The category $\mathcal A(\C)$, resp. $\mathcal B(\C)$, is the reflection of $\C$ in the category of additive categories with cokernels, resp. kernels, and cokernel, resp. kernel, preserving functors. Then the iteration $\mathcal A\mathcal B(\C)$ is the reflection of $\C$ in the category of abelian categories and exact functors. We call $\mathcal A(\C)$ and $\mathcal B(\C)$ the Freyd categories of $\C$ since the first systematic study of these categories was done by Freyd in the mid-sixties. The purpose of the paper is to study further the Freyd categories and to indicate their applications to the module theory of an abelian or triangulated category.

It is proved that equalities between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equality in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categories, which is related to model-theoretical methods of normalization., 8 pages

This chapter focuses on the syntactic categories. In one alternative view, categories have a prototype structure, which entails the following two claims for linguistic theory: Categorial Prototypicality: a. Grammatical categories have "best case" members and members that systematically depart from the "best case". b. The optimal grammatical description of morphosyntactic processes involves reference to degree of categorial deviation from the "best case". The chapter presents approaches that descriptively adequate grammars are said to make reference to graded categorial structure. Its goal is to defend the classical theory of categories. The chapter first provides a historical background to a prototype-based approach to syntactic categories. Then, it discusses how prototype theory has been applied in this regard and discusses the major consequences that have been claimed to follow from categories having a prototype structure. The chapter provides evidence that prototypical members of a category manifest more morphosyntactic complexity than non-prototypical members. Keywords: Categorial Prototypicality; classical theory of categories; morphosyntactic complexity; prototype theory; prototype-based account; syntactic categories

Ideals are used to define homological functors for additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups. Applications are considered: derived categories and functors., Comment: 10 pages, AMS-LaTex; v.2 includes applications to derived categories and functors (preliminary version)

A basic fact in the representation theory of a semisimple Lie algebra is that the annihilator of a Verma module is generated by its intersection with the centre. This result is essential to the Beilinson-Bernstein equivalence of categories and to the classification of the primitive spectrum. The usual proof is deceptively simple since it is based on a deep This result also leads quickly to Kostant’s fundamental separation theorem. Recently Bernstein and Lunts gave a very simple proof of the separation theorem; but this did not appear to recover multiplicities, nor Verma module annihilators. However, multiplicities can be recovered through the work of Hesselink. Moreover, by exploiting a technique developed in conjunction with G. Letzter one obtains Verma module annihilators. In principle, this makes possible a development of primitive ideal theory in more general situations.

This chapter discusses derived categories and their uses. Derived categories are a formalism for hyperhomology. Derived categories found their first applications in duality theory in the coherent setting, in etale, and in the locally compact setting. Beilinson and Bernstein and Gelfand used derived categories to establish a relation between coherent sheaves on projective space and representations of certain finite-dimensional algebras. Morita theory has further widened the range of applications of derived categories. Broue's conjectures on representations of finite groups are typical of the synthesis of precision with generality that can be achieved by the systematic use of this language. Any relation formulated in the language of derived categories and functors gives rise to assertions formulated in the more traditional language of cohomology groups, filtrations, and spectral sequences. Heller's construction is also presented in the chapter. Heller's stable categories provide an efficient approach to the homotopy category. They also yield many other important examples of triangulated categories and of suspended categories.

We use a new description of Kachel semigroup as triples with a pushout product. A simple Clifford category as a canonical representative (under the relation of equivalence of categories) of the standard Clifford category of this semigroup is considered.

In Sect. 2.3 we have noted that modules over a division algebra D and modules over the simple algebra M n (D) are “equally structured”. Results of Sect. 2.6 show that, in general, modules over isotypic semisimple algebras possess the same properties: such modules have isomorphic endomorphism rings, etc. In Sect. 3.5 these results have been extended to projective modules over arbitrary isotypic algebras (Lemma 3.5.5). It turns out that one can remove the requirement of projectivity: All modules over isotypic algebras are equally structured. However, in order to formulate this statement properly, it is necessary to introduce a number of concepts which presently play an important role in various areas of mathematics. Above all, it is the concept of a category and a functor, as well as the notion of an equivalence of categories, which appears to be a mathematical formulation of the expression “equally structured”.

For many of the applications of K-theory, it is useful to have the notion of K-theory for categories and not just for rings. In this more general context, the K-theory of a ring Ris just the K-theory of the category Proj R of finitely generated projective modules over R. Another natural example is the topological K-theory of a compact space X, which is the K-theory of the category Vect X of (locally trivial, real, or complex) vector bundles over X. The identification of this with the K-theory of the ring R= C(X) then follows from an equivalence of categories Proj R ≅ Vect X, But there axe also many examples that don’t come so directly from rings; for instance, if X is a projective algebraic variety, one can consider in a similar way the category Vect Xof algebraic vector bundles over X.We will see many more examples shortly

Dade [D1, Theorem 7.4] obtained an important result on the equivalence of categories, extending the classical stable Clifford theory. He used the theory of strongly graded rings. Recently, this work has been generalized to arbitrary graded rings, see E. Dade [D2], [D3], J.L. Gomez Pardo and C. Nǎstǎsescu [GN ], C. Nǎstǎsescu and F. Van Oystaeyen [NVO2]. In the classical case the stable Clifford theory relates isomorphism classes of simple modules on a strongly graded ring R which are direct sums of a fixed simple Re-module, where Re is the component of degree e, with the isomorphism classes of simple modules on a crossed product. The aim of this paper is extend the foregoing result to C-cocritical modules, where C is a localizing subcategory, on divisorially graded rings. We start with a relative version of Clifford theory using the simple objects of the quotient category. We investigated the situation of the so-called divisorially graded rings introduced by F. Van Oystaeyen in the commutative case and then generalized by many other author to more general situations (see the monograph [LVVO] and the references quoted there). We will work in the categories of R-Mod and R − gr, thus we prefer use the a general Grothendieck category and the concept of static objects in this kind of category to establish our basic results. The paper is organized as follows. After a Section of preliminaries, we introduce the notion of static objects in quotient categories in the next section. If we have adjoints functors between two Grothendieck categories A and B and a localizing subcategory C of A, then we show that under certain conditions it is possible to obtain an equivalence between the category of static objects inA/C and the category of co-static objects of some quotient category of B. In the last Section, we apply

Four different alternatives for the definition of standard equational algebraic specifications and the corresponding specification morphisms are studied and compared. Although intuitively the definitions may appear equivalent, they lead to three different equivalence classes of categories. It is shown, in fact, that the construction of pushouts and pullbacks is significantly different in these three cases. Although the corresponding specification logics are all semantical equivalent in a weak sense, three of them are semantical inconsistent with respect to pullback constructions. The nonequivalence of the equational categories has also significant implications for the corresponding high-level-replacement (HLR) system.

In many situations, minimal models are used as representatives of homotopy types. In this paper we state this fact as an equivalence of categories . This equivalence follows from an axiomatic definition of minimal objects. We see that this definition includes examples such as minimal resolutions of Eilenberg-Nakayama-Tate, minimal fiber spaces of Kan and A-minimal A-extensions of Halperin . For the first one, this is done by generalizing the construction of minimal resolutions of modules to complexes. The others follow by a caracterization of minimal objects in bifibred categories.

Recent studies on the ways in which we form categories of objects indicate that we admit that there are special elements (prototypes) in most classes. Moreover, these classes of objects seem to be organized in such a way that apparently it exists a special hierarchical level, the basic level, that is generally used in communication. Researchers have identified several types of cognitive models that seem to play an important role in the ways in which we form and organize these categories. The explanatory power of these models is tested in the reinterpretation of findings from several investigations.

We construct a functor from the category of modules over the trigonometric (resp., rational) Cherednik algebra of type gl(l) to the category of integrable modules of level l over a Yangian for the loop algebra sl(n) (resp., over a subalgebra of this loop Yangian) and we establish that it is an equivalence of categories if l+2 < n. Finally, we treat the case of the rational Cherednik algebras of type A(l-1).

This paper proves Brown Representability for stable categories. These categories are obtained from additive categories by declaring all morphisms which factor through objects from a fixed pre-enveloping class to be zero. Brown Representability for such categories is obtained by first proving that stable categories are what Brown in [2] calls ‘‘abstract homotopy categories’’, next using Brown’s theory for these categories. The result is applied to two cases: one is a category of complexes of modules over a ring. Here we recover a representability theorem for functors on derived categories, first given by Neeman in [11, Thm. 3.1]. The other case is the category of modules over an Artinian ring. Here we obtain a result on representability, Theorem 4.4, which appears to be new.

Various familiar concepts of algebraic categories (such as primitive and quasiprimitive categories of algebras as well as varietal, monadic, regular monadic, and regular categories) are contrasted with each other and with the new notion of essentially algebraic categories. The new notion has pleasant features and covers several cases of an apparently algebraic nature, which are not covered by any of the older concepts.

It is long known that Lawvere's theory in The category of categories as foundations of mathematics A[1] does not work, as indicated in Ishell's review [0]. Isbell there gives a counterexample that CDT—Category Description Theorem—[1, p. 15] is in fact not a theorem of BT (the Basic Theory of [1]) and suggests adding CDT to the axioms.Our starting point was the claim in [1] that “the basic theory needs no explicit axiom of infinity.” We define a model ℳ of BT in which all categories are finite. In particular, the “monoid of nonnegative integers N” coincides in ℳ with the terminal object 1. We study ℳ in some detail in order to establish the true status of various “theorems” or “metatheorems” of BT: The metatheorem of [1, p. 11] saying that the discrete categories form a category of sets, CDT, the theorem on p. 15, and the theorem on p. 16 of [1] are all nontheorems. The remaining results indicated in [1] concerning BT are provable. However, as the Predicative Functor Construction Schema—PFCS—are justified in [1] by using the “metatheorem” and CDT, we provide a proof of these two schemata by showing that the discrete categories of BT (or of convenient extensions of BT) form a two-valued Boolean topos.

We show that if S is a cocontinuous triple on a full reflective subcategory of a functor category then the category of S-algebras is again a full reflective subcategory of a functor category. This note should be considered an addendum to [41, and definitions for all of the terminology and concepts we use can be found there. We shall also fix V to be a closed bicomplete category and, in this note, all of the category theory is done relative to V. In [41, we have shown the following: (I) If C is a small category and T is a cocontinuous triple on the functor category VC, then there is a small category C' and a functor /: C -, C' so that (a) T is the triple induced by the adjoint pair (/*, /): VC' VC, where /*: Vc ) Vc is the functor induced by / and /1 is the left adjoint of / *; (b) the adjoint pair (/ *, /1) is tripleable, so that there is an equivalence of categories vC (VC)T, where (VC)T is the category of Talgebras. (II) If C is a small category and T is any triple on Vc, there is a unique cocontinuous triple T on V0 and a map of triples r: T so that, if R: Co )VC denotes the right Yoneda embedding of C into the representable functors of VC, rR is the identity (we shall refer to T as the cocontinuous approximation to T). In this paper, we shall prove the following 1. Theorem. Suppose C is a small category, A is a full reflective subcategory of VC (i.e. the inclusion functor of A to VC has a left adjoint) and S is a cocontinuous triple on A. Then there is a small categoPresented to the Society January 25, 1973; received by the editors August 16, 1972 and, in revised form, August 6, 1973. AMS(MOS) subject classifications (1970). Primary 18C15; Secondary 18E99, 18F 20. Copyright C 1975, American Mathematical Societv

The concept of finitely presented functor was introduced by Auslander. Proposition 3.1 of Auslander & Reiten provides a way of dealing with the category of finitely presented functors, that seems concrete and easy to use, at least in some examples. The study of this category, using this particular line of thought, is the main purpose of this work. In §1 I recall some basic definitions and give the required notation. In §2 I state the theorem of Auslander & Reiten referred to above and give a new proof of this result. The first part of this proof is an immediate consequence of the theory developed by Green. In §3 I state and prove an unpublished theorem by J. A. Green and I introduce a new category I such that the category of finitely presented functors. mmod A , is equivalent to a quotient category I / J , where J is an ideal of I . In §4 I give some examples of properties of mmod A , stated and proved in terms of the category I , by using the equivalence of categories referred to in §3. In §5 I consider the particular case where A = A q = k -alg < z : z q = 0>, apply the results of previous sections to study mmod A q and make conclusions about the representation type of the Auslander algebra of A q .

We construct derived functors in additive categories in which each morphism has a kernel, co-kernel, image, and coimage, but the image and coimage are not necessarily isomorphic. We prove that these derived functors possess the usual properties. The main difficulty is that the 3×3-lemma does not necessarily hold in the categories under consideration.

Consider an adjunction : K. → A, T = the monad it induces in K and o: A → KT the comparison functor, KT being the category of T-algebras. By o*: Proj U → Proj UT we denote the restriction and co-restriction of o to the subcategories of U-projective and UT -projective objects, respectively. In this paper we deal with the following problem, raised by R.-E. Hoffmann in [5] 1.16 (b): Assuming that o* is an equivalence of categories when is it possible to find a category C and a right adjoint functor V: C → K inducing the same monad T in K, and a full reflective embedding E: A → K, such that: (1) V.E = U. (2) o = o'. E for the comparison functor o': C → KT . (3) F'X is contained (via E) in A, for each K-object X, F' being the left adjoint of V. (4) o': C → KT has a full and faithful left adjoint L'. We prove that there exists a pair (C,V) satisfying the conditions of the problem, with A an isomorphism-closed subcategory of C, such that: (5) For all C ⊂ Obj C the reflection map rC: C →...

This note is a complement to a beautiful recent paper of A. Carboni, which characterizes affine categories, i.e. those categories which occur as slices of additive categories. We show that the condition that every reflexive graph has a unique groupoid structure, which was observed by Carboni to follow from affineness, is equivalent to the existence of a natural Mal'cev operation on a category; we further show that this condition implies the additiveness of the category of pointed objects, but not the affineness of the original category.

The structure of E-unitary inverse semigroups has been described by McAlister and by Reilly and the author. The parameters in the first structure theorem may be made into a category, and the same holds for the parameters in the second structure theorem. We prove that each of these categories is equivalent to the category of E-unitary inverse semigroups and their homomorphisms. We also provide functors between the two first-mentioned categories which are naturally equivalent to the composition of the functors figuring in the categorical equivalence referred to above.

We generalize the Morita theorems to certain functor categories using properties of adjoint functors. Let v1 and v be categories. Recall that z1 and X are said to be equivalent if there are functors u: and v: Vd1 and natural isomorphisms of functors -q: I > vu and e: uv => I', where I and I' denote the identity functors on Q/ and X respectively. When & = _R and 6= 4, where R and S are unitary rings and A and As denote the categories of right R-modules and right S-modules respectively, the Morita theorems state that the pairs of functors (v, u), u: -q and v: X Z, which define an equivalence of categories between v1 and X are in one-to-one correspondence with bimodules RES which are R-S progenerators, the correspondence being given by isomorphisms uOR E and v Homs (E, ) (see [2], or [1, Chapter II, ?3] for a discussion of these theorems). In general, the isomorphisms u= OR E and v Homs (E, ) give a one-to-one correspondence between bimodules RES and adjoint pairs (v, u) from s Secondary 18G99.

In this paper we shall consider a non-additive category of A-modules, that is, instead of a ring A we take a monoid A which acts on sets from the left. These objects will be called A-acts. We investigate indecomposable A-acts and generators and characterize projectives in this category. For a given monoid A we describe all monoids B such that the category of B-acts is equivalent to the category of A-acts. In particular we find that equivalence of these categories yields an isomorphism between the monoids A and B if A is a group or finite or commutative. This differs from the additive case where the categories of modules over a commutative field and its ring of nxn matrices are equivalent. Finally we give examples of non-isomorphic monoids A and B such that the corresponding categories are equivalent.