Finite Coproducts Research Articles

Internal Neighbourhood Structures II: Closure and closed morphisms

Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in \cite{2020}. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces analogues of several well-known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit that preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of \emph{continuity} of morphisms with respect to induced closure operators.

Finite coproducts in the category of quadratic modules of Lie algebras

In this study, we will construct finite coproduct objects in the category of quadratic modules of Lie algebras with a new approach using the idea of quasi-quadratic modules.

The stable category of preorders in a pretopos I: General theory

In a recent article Facchini and Finocchiaro considered a natural pretorsion theory in the category of preordered sets inducing a corresponding stable category. In the present work we propose an alternative construction of the stable category of the category $\mathsf{PreOrd} (\mathbb C)$ of internal preorders in any coherent category $\mathbb C$, that enlightens the categorical nature of this notion. When $\mathbb C$ is a pretopos we prove that the quotient functor from the category of internal preorders to the associated stable category preserves finite coproducts. Furthermore, we identify a wide class of pretoposes, including all $\sigma$-pretoposes and all elementary toposes, with the property that this functor sends any short $\mathcal Z$-exact sequences in $\mathsf{PreOrd} (\mathbb C)$ (where $\mathcal Z$ is a suitable ideal of trivial morphisms) to a short exact sequence in the stable category. These properties will play a fundamental role in proving the universal property of the stable category, that will be the subject of a second article on this topic.

(Co)Limit calculations in the category of 2-crossed $R$-modules

In this work, we obtain how to construct finite limits and colimits for 2-crossed $R$-Modules over groups denoted with $\mathbf{X_2Mod/R}$. We give direct construction of the pullback object to show that this category has finite products over the terminal object. We also show finite coproducts and (co)completeness.

On (co)products of partial combinatory algebras, with an application to pushouts of realizability toposes

AbstractWe consider two preorder-enriched categories of ordered partial combinatory algebras: OPCA, where the arrows are functional (i.e., projective) morphisms, and OPCA†, where the arrows are applicative morphisms. We show that OPCA has small products and finite biproducts, and that OPCA† has finite coproducts, all in a suitable 2-categorical sense. On the other hand, OPCA† lacks all nontrivial binary products. We deduce from this that the pushout, over Set, of two nontrivial realizability toposes is never a realizability topos. In contrast, we show that nontrivial subtoposes of realizability toposes are closed under pushouts over Set.

Internal coalgebras in cocomplete categories: Generalizing the Eilenberg–Watts theorem

The category of internal coalgebras in a cocomplete category [Formula: see text] with respect to a variety [Formula: see text] is equivalent to the category of left adjoint functors from [Formula: see text] to [Formula: see text]. This can be seen best when considering such coalgebras as finite coproduct preserving functors from [Formula: see text], the dual of the Lawvere theory of [Formula: see text], into [Formula: see text]: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of [Formula: see text] into [Formula: see text]. Since [Formula: see text]-coalgebras in the variety [Formula: see text] for rings [Formula: see text] and [Formula: see text] are nothing but left [Formula: see text]-, right [Formula: see text]-bimodules, the equivalence above generalizes the Eilenberg–Watts theorem and all its previous generalizations. By generalizing and strengthening Bergman’s completeness result for categories of internal coalgebras in varieties, we also prove that the category of coalgebras in a locally presentable category [Formula: see text] is locally presentable and comonadic over [Formula: see text] and, hence, complete in particular. We show, moreover, that Freyd’s canonical constructions of internal coalgebras in a variety define left adjoint functors. Special instances of the respective right adjoints appear in various algebraic contexts and, in the case where [Formula: see text] is a commutative variety, are coreflectors from the category [Formula: see text] into [Formula: see text].

Ultrafilters, finite coproducts and locally connected classifying toposes

We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin–Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF. The result in its most basic form states that the category FC(Set,Set) of finite-coproduct-preserving endofunctors of Set is equivalent to the presheaf category [UF,Set]. Using this result, and some of its evident generalisations, we re-find in a natural manner the important model-theoretic realisation relation between n-types and n-tuples of model elements; and draw connections with Makkai and Lurie's work on conceptual completeness for first-order logic via ultracategories.As a further application of our main result, we use it to describe a first-order analogue of Jónsson and Tarski's canonical extension. Canonical extension is an algebraic formulation of the link between Lindenbaum–Tarski and Kripke semantics for intuitionistic and modal logic, and extending it to first-order logic has precedent in the topos of types construction studied by Joyal, Reyes, Makkai, Pitts, Coumans and others. Here, we study the closely related, but distinct, construction of the locally connected classifying topos of a first-order theory. The existence of this is known from work of Funk, but the description is inexplicit; ours, by contrast, is quite concrete.

Internal neighbourhood structures

The main aim of this paper is to provide a description of neighbourhood operators in finitely complete categories with finite coproducts and a proper factorisation system such that the semilattice of admissible subobjects make a distributive complete lattice. The equivalence between neighbourhoods, Kuratowski interior operators and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhoods is shown to be topological. Regular epimorphisms of categories of neighbourhoods are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the category of internal neighbourhoods of topological spaces is the category of bitopological spaces, while in the category of locales every locale comes equipped with a natural internal topology.

AMALGAMABLE DIAGRAM SHAPES

AbstractA category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram has a cocone. We show that for a finitely generated category I, the following are equivalent: (i) every I-shaped diagram in a category with the AP and the JEP has a cocone; (ii) every I-shaped diagram in the category of sets and injections has a cocone; (iii) a certain canonically defined category ${\cal L}\left( {\bf{I}} \right)$ of “paths” in I has only idempotent endomorphisms. When I is a finite poset, these are further equivalent to: (iv) every upward-closed subset of I is simply-connected; (v) I can be built inductively via some simple rules. Our proof also shows that these conditions are decidable for finite I.

REPRESENTING CONJUNCTIVE DEDUCTIONS BY DISJUNCTIVE DEDUCTIONS

AbstractA skeleton of the category with finite coproducts${\cal D}$ freely generated by a single object has a subcategory isomorphic to a skeleton of the category with finite products ${\cal C}$ freely generated by a countable set of objects. As a consequence, we obtain that ${\cal D}$ has a subcategory equivalent with ${\cal C}$. From a proof-theoretical point of view, this means that up to some identifications of formulae the deductions of pure conjunctive logic with a countable set of propositional letters can be represented by deductions in pure disjunctive logic with just one propositional letter. By taking opposite categories, one can replace coproduct by product, i.e., disjunction by conjunction, and the other way round, to obtain the dual results.

When coproducts are biproducts

AbstractAmong monoidal categories with finite coproducts preserved by tensoring on the left, we characterise those with finite biproducts as being precisely those in which the initial object and the coproduct of the unit with itself admit right duals. This generalises Houston's result that any compact closed category with finite coproducts admits biproducts.

Coproduct of 2-crossed modules: applications to a definition of a tensor product for 2-crossed complexes

In this paper it is shown how to construct finite coproducts in the category of 2-crossed modules. We give explicit descriptions of the pullback functor and induced functor along a morphism of precrossed modules which are used to built the coproduct of 2-crossed modules. As an application we give a definition and construction of a possible tensor product functor for 2-crossed complexes.

Gödel spaces and perfect MV-algebras

The category of Gödel spaces GS (with strongly isotone maps as morphisms), which are dually equivalent to the category of Gödel algebras, is transferred by a contravariant functor H into the category MV(C)G of MV-algebras generated by perfect MV-chains via the operators of direct products, subalgebras and direct limits. Conversely, the category MV(C)G is transferred into the category GS by means of a contravariant functor P. Moreover, it is shown that the functor H is faithful, the functor P is full and the both functors are dense. The description of finite coproduct of algebras, which are isomorphic to Chang algebra, is given. Using duality a characterization of projective algebras in MV(C)G is given.

Finitary -adhesive categories

Finitary$\mathcal{M}$-adhesive categories are$\mathcal{M}$-adhesive categories with finite objects only, where$\mathcal{M}$-adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of$\mathcal{M}$-subobjects. In this paper, we show that in finitary$\mathcal{M}$-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for$\mathcal{M}$-adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary$\mathcal{M}$-adhesive categories have a unique$\mathcal{E}$-$\mathcal{M}$factorisation and initial pushouts, and the existence of an$\mathcal{M}$-initial object implies we also have finite coproducts and a unique$\mathcal{E}$′-$\mathcal{M}$pair factorisation. Moreover, we can show that the finitary restriction of each$\mathcal{M}$-adhesive category is a finitary$\mathcal{M}$-adhesive category, and finitarity is preserved under functor and comma category constructions based on$\mathcal{M}$-adhesive categories. This means that all the classical results are also valid for corresponding finitary$\mathcal{M}$-adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-$\mathcal{M}$-adhesive categories.

Composites of Central Extensions Form a Relative Semi-Abelian Category

We consider trivial and central extensions, in the sense of G. Janelidze and G. M. Kelly, which are defined with respect to an adjunction between a Barr-exact category C and a Birkhoff subcategory X of C. Assuming in addition that C is a pointed Mal’tsev category with cokernels, and that X is protomodular, we prove that: (a) the class of all trivial extensions and the class of all finite composites of central extensions form relative homological category structures on C; (b) if C has finite coproducts, then the class of all finite composites of central extensions forms a relative semi-abelian category structure on C.

A Colimit Decomposition for Homotopy Algebras in Cat

Badzioch showed that in the category of simplicial sets each homotopy algebra of a Lawvere theory is weakly equivalent to a strict algebra. In seeking to extend this result to other contexts Rosický observed a key point to be that each homotopy colimit in SSet admits a decomposition into a homotopy sifted colimit of finite coproducts, and asked the author whether a similar decomposition holds in the 2-category of categories Cat. Our purpose in the present paper is to show that this is the case.

Representing Geometric Morphisms Using Power Locale Monads

It is shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. The adjunctions are characterized as those that preserve the order enrichment, commute with the double power locale monad and whose right adjoints preserve finite coproduct. They are also characterized as those adjunctions that preserve the order enrichment and commute with both the lower and the upper power locale monads.

From Coalgebraic to Monoidal Traces

The main result of this paper shows how coalgebraic traces, in suitable Kleisli categories, give rise to traced monoidal structure in those Kleisli categories, with finite coproducts as monoidal structure. At the heart of the matter lie partially additive monads inducing partially additive structure in their Kleisli categories. By applying the standard “Int” construction one obtains compact closed categories for “bidirectional monadic computation”.

Coherence for monoidal monads and comonads

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal monads and comonads, that is, monads and comonads in a monoidal category such that the endofunctor of the monad or comonad is a monoidal functor (this means that it preserves the monoidal structure up to a natural transformation that need not be an isomorphism). These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. The monoidal structure is also allowed to be given with finite products or finite coproducts. Monoidal comonads with finite products axiomatise a plausible notion of equality of deductions in a fragment of the modal logic S4.

Boolean and classical restriction categories

A restriction category is an abstract category of partial maps. A Boolean restriction category is a restriction category that supports classical (Boolean) reasoning. Such categories are models of loop-free dynamic logic that is deterministic in the sense that < α > Q ⊂ [α]Q. Classical restriction categories are restriction categories with a locally Boolean structure: it is shown that they are precisely full subcategories of Boolean restriction categories. In particular, a Boolean restriction category may be characterised as a classical restriction category with finite coproducts in which all restriction idempotents split.Every restriction category admits a restriction embedding into a Boolean restriction category. Thus, every abstract category of partial maps admits a conservative extension that supports classical reasoning. An explicit construction of the classical completion of a restriction category is given.