Flatness, weakly lex colimits, and free exact completions

We capture in the context of lex colimits, introduced by Garner and Lack, the universal property of the free regular and Barr-exact completions of a weakly lex category. This is done by introducing a notion of flatness for functors $F\colon\mathcal C\to\mathcal E$ with lex codomain, and using this to describe the universal property of free $\Phi$-exact completions in the absence of finite limits, for any given class $\Phi$ of lex weights. In particular, we shall give necessary and sufficient conditions for the existence of free lextensive and free pretopos completions in the non-lex world, and prove that the ultraproducts, in the categories of models of such completions, satisfy an universal property

Flat vs. filtered colimits in the enriched context

The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as $\mathbf{Ab},\mathbf{SSet},\mathbf{Cat}$ and $\mathbf{Met}$. The problem in this context is that the equivalence between (a) and (b) is no longer true in general. The aim of this paper is then to: (1) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow$ (b) holds; (2) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow$ (b) holds up to Cauchy completion; (3) explore some examples not covered by (1) or (2).Comment: Revised version: major changes to the introduction, added some words at the beginning of Sect. 3 and 4. To appear on Advances in Mathematic

Enriched universal algebra

Following the classical approach of Birkhoff, we introduce enriched universal algebra. Given a suitable base of enrichment $\mathcal V$, we define a language $\mathbb L$ to be a collection of $(X,Y)$-ary function symbols whose arities are taken among the objects of $\mathcal V$. The class of $\mathbb L$-terms is constructed recursively from the symbols of $\mathbb L$, the morphisms in $\mathcal V$, and by incorporating the monoidal structure of $\mathcal V$. Then, $\mathbb L$-structures and interpretations of terms are defined, leading to enriched equational theories. In this framework we prove several fundamental theorems of universal algebra, including the characterization of algebras for finitary monads on $\mathcal V$ as models of an equational theories, and several Birkhoff-type theorems

Accessible categories with a class of limits

In this paper we characterize those accessible $\mathcal V$-categories that have limits of a specified class. We do this by introducing the notion of companion $\mathfrak C$ for a class of weights $\Psi$, as a collection of special types of colimit diagrams that are compatible with $\Psi$. We then characterize the accessible $\mathcal V$-categories with $\Psi$-limits as those accessibly embedded and $\mathfrak C$-virtually reflective in a presheaf $\mathcal V$-category, and as the $\mathcal V$-categories of $\mathfrak C$-models of sketches. This allows us to recover the standard theorems for locally presentable, locally multipresentable, and locally polypresentable categories as instances of the same general framework. In addition, our theorem covers the case of any weakly sound class $\Psi$, and provides a new perspective on the case of weakly locally presentable categories.Comment: Journal version. Some references added, as well as 4.10, 4.11, 4.18, 4.20(4

Strongly Preserved Formulas in Topoi

Topoi originated in the 1960's when Grothendieck found a powerful way to study categories related to algebraic geometry; in its idea every topological space gives a topos, namely the category of sheaves on the space. These are now known as ``Grothendieck topoi'' and constitute a particular case of ``elementary topoi'', introduced by Lawvere and Tierney in 1968-69. The biggest problem of Grothendieck topoi, in Lawvere idea, was their extremely complex constructs; thus he tried to describe the most relevant aspect of topoi in a much more simple way. Essentially, a topos is a category with a terminal object, all finite products, exponential objects and a subobject classifier. The few properties defining a topos permit us to work with (first and higher order) logic and, in particular, model theory. Given a language L, we define the notion of L-structure in a topos simply miming the usual construction in Sets. Moreover, we can define the interpretation of a formula as a particular subobject of the topos (here we follow ``Sketches of an Elephant''). This allows us to say when a formula is true (i.e. the top element in the Heyting algebra of subobjects) or false (i.e. the bottom element); thus we can introduce the notion of model for a given theory T and hence build the category of T-models in a topos E (with a suitable notion of homomorphism). In this context, a question arises: which formulas are ``preserved'' by morphism between topoi? Or even: are there some kind of theories whose models are ``preserved'' by geometric morphisms? First we need to clarify what we mean by preservation; we say that a formula is strongly preserved if for each geometric morphism f and each structure in a topos, the subobjects identified by the formula coincides under the inverse image of f. A class of strongly preserved first-order formulas is easily identified; it is the smallest containing atomic formulas and closed for conjunction, disjunction and existential quantification; these are known as geometric formulas. Following Blass' paper ``Fixed Point Preservation'', we can consider a new construction and prove that the class of strongly preserved formulas goes beyond first-order. We call the elements of this class EFPL-formulas. As a consequence, we are able to prove the following: Theorem: Let T be a theory whose axioms are sequents of EFPL-formulas; then for each geometric morphism f, its inverse image induces a functor between the categories of T-models. Finally, we talk about classifying topoi for theories; initially we get a result for Grothendieck topoi and geometric theories: Theorem: Let T be a geometric theory; then there exists a Grothendieck topos which classifies T-models. To talk about classifying topoi for EFPL theories we need a difference approach: the equivalence we want to prove will be no more for Grothendieck topoi, but for elementary ones over a base topos. The only weaker hypothesis we must subsume are some requests of finiteness and the existence of a natural number object in the base topos (i.e. a model of natural numbers): Theorem: Let L be a first-order language with a finite number of dorts and symbols, T be a finite theory whose axioms are sequents of strongly preserved formulas and E be a topos with a natural number object. Then T has a classifying topos over E. To conclude, as shown in ``Classifying topoi and the axiom of infinity'', we prove that the request of a natural number object in the last theorem is necessary

Enriched regular theories

Empirical thesis.Bibliography: pages 51-52.Introduction -- 1. Background -- 2. Bases for enrichment -- 3. Regular and exact V-categories -- 4. Definable V-categories -- 5. Future directions -- Bibliography.Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. In 1986 Barr showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosicky, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Considering a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosicky.Mode of access: World wide web1 online resource (i, 52 pages

Topics in the theory of enriched accessible categories

The aim of this thesis is to further develop the theory of accessible categories in the enriched context. We study and compare the two notions of accessible and conically accessible -categories, both arising as free cocompletions of small -categories: the former under flat-weighted colimits and the latter under filtered colimits. These two notions are not the same in general, however we show that they coincide for many significant bases of enrichment such as Cat and SSet, and differ just by Cauchy completeness for many algebraic examples including Ab, R-Mod and GAb. We then provide new characterization theorems for these by considering some notions of virtual orthogonality and virtual reflectivity which generalize the usual reflectivity and orthogonality conditions for locally presentable categories. The word virtual refers to the fact that the reflectivity and orthogonality conditions are given in the free completion of the -category involved under small limits, instead of the -category itself. We then prove that the 2-category of accessible -categories, accessible -functors, and -natural transformations has all flexible limits. In the final chapters we study, characterize, and provide duality theorems in the setting of accessible -categories with limits of a specified class Ψ; in this context, instead of the free completion under small limits, we consider “free completions” under a specific type of colimits ℭ for which, in particular, ℭ-colimits commute in with Ψ-limits. This allows us to capture the theories of weakly locally presentable, locally multipresentable, locally polypresentable, and accessible categories as instances of the same general framework.</p