How accessible are categories of algebras?

For locally finitely presentable categories it is well known that categories of F-algebras, where F is a finitary endofunctor, are also locally finitely presentable. We prove that this generalizes to locally finitely multipresentable categories. But it fails, in general, for finitely accessible categories: we even present an example of a strongly finitary functor F (one that preserves finitely presentable objects) whose category of F-algebras is not finitely accessible. On the other hand, categories of F-algebras are proved to be ω₁-accessible for all strongly finitary functors—and it is an open problem whether this holds for all finitary functors.     

Lawvere theories enriched over a general base

We generalise the correspondence between Lawvere theories and finitary monads on in two ways. First, we allow our theories to be enriched in a category V that is locally finitely presentable as a symmetric monoidal closed category: symmetry is convenient but not necessary. And second, we allow the arities of our theories to be finitely presentable objects of a locally finitely presentable V-category A. We call the resulting notion that of a Lawvere A-theory. We extend the correspondence for ordinary Lawvere theories to one between Lawvere A-theories and finitary V-monads on A. We illustrate this with examples leading up to that of the Lawvere -theory for cartesian closed categories, i.e., the -enriched theory on the category for which the models are all small cartesian closed categories. We also briefly investigate change-of-base.     

An embedding theorem for regular Mal'tsev categories

In this paper, we obtain a non-abelian analogue of Lubkin's embedding theorem for abelian categories. Our theorem faithfully embeds any small regular Mal'tsev category $\mathbb{C}$ in an n-th power of a particular locally finitely presentable regular Mal'tsev category. The embedding preserves and reflects finite limits, isomorphisms and regular epimorphisms, as in the case of Barr's embedding theorem for regular categories. Furthermore, we show that we can take n to be the (cardinal) number of subobjects of the terminal object in $\mathbb{C}$.     

Bounds for norms of the matrix inverse and the smallest singular value

In the first part, we obtain two easily calculable lower bounds for ‖A^-1‖, where ‖·‖ is an arbitrary matrix norm, in the case when A is an M-matrix, using first row sums and then column sums. Using those results, we obtain the characterization of M-matrices whose inverses are stochastic matrices. With different approach, we give another easily calculable lower bounds for ‖A^-1‖_∞ and ‖A^-1‖₁ in the case when A is an M-matrix. In the second part, using the results from the first part, we obtain our main result, an easily calculable upper bound for ‖A^-1‖₁ in the case when A is an SDD matrix, thus improving the known bound. All mentioned norm bounds can be used for bounding the smallest singular value of a matrix.     

Cocomplete toposes whose exact completions are toposes

Let E be a cocomplete topos. We show that if the exact completion of E is a topos then every indecomposable object in E is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere-Schanuel characterization of Boolean presheaf toposes and Hofstra’s characterization of the locally connected Grothendieck toposes whose exact completion is a Grothendieck topos.We also show that for any topological space X, the exact completion of is a topos if and only if X is discrete. The corollary in this case characterizes the Grothendieck toposes with enough points whose exact completions are toposes.     

Flat covers in abelian and in non-abelian categories

A general existence theorem for flat covers in (e.g., quasi-abelian) locally finitely presented categories is obtained from an additive Ramsey type theorem. In the abelian case, it is shown that flat covers always exist. Applications to categories of separated presheaves or sheaves, localizations of Bousfield type, torsion-free classes of finite type, and categories of filtered objects or complexes, are given.     

A contribution to the closed geodesic problem

The closed geodesic problem has been solved by many authors under additional hypothesis. In this paper we develop a new way to solve this problem, by defining finitely many obstruction classes. This permits to enlarge significantly the family of manifolds for which this problem is solved. Our method which relies on the concept of A_∞-section of a homomorphism of differential graded algebras, shows explicitly how the natural structure of shc-algebra on the singular cochains of a space comes into play.     

Set-theoretic constructions of nonshrinking open covers

A family {M_α|α?A} is a shrinking of a cover {O_α|α?A} of a topological space if {M_α|α?A} also covers and M_α?O_α for all α?A.?⁺⁺ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking.? implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [17] to show that any space which has a normal product with every P-space is metrizable.     

The countable Telescope Conjecture for module categories

By the Telescope Conjecture for Module Categories, we mean the following claim: “Let R be any ring and (A,B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A,B) is of finite type.”We prove a modification of this conjecture with the word ‘finite’ replaced by ‘countable.’ We show that a hereditary cotorsion pair (A,B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A,B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.     

On Hopf algebra structures over free operads

The operad Lie can be constructed as the operad of primitives PrimAs from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore. We replace the operad As by families of free operads P, which include the operad Mag freely generated by a non-commutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincaré-Birkhoff-Witt type theorems and collect information about the operads PrimP, e.g., in terms of characteristic functions.     

Invariants of t-structures and classification of nullity classes

We construct an invariant of t-structures on the derived category of a commutative noetherian ring. This invariant is complete when restricting to the category of complexes with finitely generated bounded homology, and also gives a classification of nullity classes with the same restriction. On the full derived category of Z we show that the class of distinct t-structures do not form a set.     

On the normal completion of a Boolean algebra

A familiar construction for a Boolean algebra A is its normal completion, given by its normal ideals or, equivalently, the intersections of its principal ideals, together with the embedding taking each element of A to its principal ideal. In the classical setting of Zermelo-Fraenkel set theory with Choice, is characterized in various ways; thus, it is the unique complete extension of A in which the image of A is join-dense, the unique essential completion of A, and the injective hull of A.Here, we are interested in characterizing the normal completion in the constructive context of an arbitrary topos. We show among other things that it is, even at this level, the unique join-dense, or alternatively, essential completion. En route, we investigate the functorial properties of and establish that it is the reflection of A, in the category of Boolean homomorphisms which preserve all existing joins, to the complete Boolean algebras. In this context, we make crucial use of the notion of a skeletal frame homomorphism.     

Maximal parabolic regularity for divergence operators including mixed boundary conditions

We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.     

Quasi-coproducts and accessible categories with wide pullbacks

We establish a 2-categorical duality involving the 2-category A of all -accessible categories with wide pullbacks, also known as locally -polypresentable categories, and of functors preserving -filtered colimits and wide pullbacks. Commutation of wide pullbacks with so-called quasi-coproducts in Set is the basic ingredient to this duality, which leads to a full characterization of categories of type Wdpb Filt (A, Set)=A The first author acknowledges financial assistance from a special research grant of the Faculty of Arts at York University. The second author is partially supported by an NSERC operating grant.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

Coinverters and categories of fractions for categories with structure

A category of fractions is a special case of acoinverter in the 2-categoryCat. We observe that, in a cartesian closed 2-category, the product of tworeflexive coinverter diagrams is another such diagram. It follows that an equational structure on a categoryA, if given by operationsA ⁿ A forn N along with natural transformations and equations, passes canonically to the categoryA [^–1] of fractions, provided that is closed under the operations. We exhibit categories with such structures as algebras for a class of 2-monads onCat, to be calledstrongly finitary monads.The first and third authors gratefully acknowledge the support of the Australian Research Council.     

Semi-dualizing modules and related Gorenstein homological dimensions

A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom_A(C,C)?A in the derived category D(A).We show how each such module gives rise to three new homological dimensions which we call C-Gorenstein projective, C-Gorenstein injective, and C-Gorenstein flat dimension, and investigate the properties of these dimensions.     

A remark on conservative cocompletions of categories

For a set F of small categories, F-conservative cocompletions of a category are cocompletions preserving all existing colimits of type F. We prove that every category has a free F-conservative cocompletion. However, unless F is trivial, this cocompletion fails in general to be locally small.     

Analysis of a problem of Raikov with applications to barreled and bornological spaces

Raikov’s conjecture states that semi-abelian categories are quasi-abelian. A first counterexample is contained in a paper of Bonet and Dierolf who considered the category of bornological locally convex spaces. We prove that every semi-abelian category I admits a left essential embedding into a quasi-abelian category K_l(I) such that I can be recovered from K_l(I) by localization. Conversely, it is shown that left essential full subcategories I of a quasi-abelian category are semi-abelian, and a criterion for I to be quasi-abelian is given. Applied to categories of locally convex spaces, the criterion shows that barreled or bornological spaces are natural counterexamples to Raikov’s conjecture. Using a dual argument, the criterion leads to a simplification of Bonet and Dierolf’s example.     

The Frobenius exponent of Cartier subalgebras

Let R be a standard graded finitely generated algebra over an F-finite field of prime characteristic, localized at its maximal homogeneous ideal. In this note, we prove that the Frobenius complexity of R is finite. Moreover, we extend this result to Cartier subalgebras of R.