We give a 𝒱 -enriched categorical semantics for Intuitionistic Subexponential Linear Logic (ISELL), with string-diagram and numbered equational proofs in the main text for all one-step cut-reduction schemes. A model is a 𝒱 -enriched symmetric monoidal closed category (SMCC) equipped with a family of subexponential coalgebra modalities (!^a)_a∈ (𝔼,≼ ) and, whenever a≼ b , monoidal comonad morphisms θ ^b⇒ a:!^b⇒ !^a that preserve all available comonoid structure (Δ ,e) . This preservation yields an explicit “promotion prism” identity that semantically accounts for the subexponential side-condition. We build an initial enriched syntactic model by generators-and-relations and deduce completeness. We also prove a separation result showing that models without preserving comparisons cannot validate the same promotion behaviour.
We introduce a notion of action for unitary magmas and use it to classify a broad class of split extensions, called retraction points. Our approach is based on previous work on split extensions and semidirect products of unitary magmas, extending it to the more general setting in which the middle object of a split extension need not be in bijection with the Cartesian product of the kernel and cokernel objects. The construction is based on the notion of semibiproduct rather than the classical semidirect product of groups. Although this phenomenon does not occur in groups or, more generally, in associative semiabelian varieties, it appears in monoids through weakly Schreier extensions and in certain non-associative structures such as semi-left-loops. These results provide a broader framework for the study of split extensions of unitary magmas and related categorical structures.
Abstract A $$\mathcal {C}-\text {set}$$ C - set is a functor from a category $$\mathcal {C}$$ C to the category of finite sets, and $$\mathcal {C}-\text {set}$$ C - set denotes the category of such functors with natural transformations as morphisms. In this work, we prove that $$\mathcal {C}$$ C is a groupoid if and only if $$\mathcal {C}-\text {set}$$ C - set has finitely many indecomposable objects, which is the categorical analog of transitive G -sets in classical group actions.
In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism α:M →𝒞(F,Z) to exhibit a functor Z:𝒜→𝒞 as an absolute M-weighted colimit of a functor F:ℬ→𝒞 . We also review, with short proofs, the various criteria for the weight M itself to be absolute, in the sense that any M-weighted colimit is absolute. Finally, we prove that any absolute M-weighted colimit can be viewed as a colimit weighted by an absolute weight M' .
S4-subordination algebras are a generalization of the closure algebras. In this paper, we give a topological representation for S4-subordination algebras by means of bitopological spaces ⟨ X,τ ,τ _S⟩ , where ⟨ X,τ⟩ is a Stone space and τ _S is a topology that enables the characterization of the subordination relation. We apply this bitopological representation to give a characterization of S5-subordination algebras and lattice subordinations. We also show that there exists a bijective correspondence between congruence compatible with the subordination and certain closed subsets of the Stone space ⟨ X,τ⟩ that are also saturated sets of the space ⟨ X,τ _S⟩ . Additionally, we explore two types of morphisms between S4-subordination algebras: one based on Boolean homomorphisms and another based on meet-homomorphisms. Finally, we provide a topological representation for each type of morphism.
We show that the fundamental category of a directed space decomposed into stellar strata by a poset (partially ordered set) is equivalent to its face category. As a consequence, we compute the fundamental category for two stellar stratifications on the classifying space of a small category.
We give an elementary characterization of those quantaloids $$\mathcal {Q}$$ Q for which the category $$\textsf{Cat}(\mathcal {Q})$$ Cat ( Q ) of $$\mathcal {Q}$$ Q -enriched categories and functors is cartesian closed. We then unify several known cases (previously proven using ad hoc methods) and we give some new examples.
We analyze the relationship between coextensive varieties, central elements, and the Gaeta topos. Although the Gaeta topos captures a natural topos-theoretic notion of indecomposability, it does not in general classify all directly indecomposable algebras. We show that, in coextensive varieties, this discrepancy has a structural explanation: the geometric content of the Gaeta topos corresponds precisely to the absence of non-trivial central elements. Using the decomposition term available in every coextensive variety, we prove that central-freeness is expressible in the original algebraic language, yielding a geometric theory stable under interpretation in arbitrary toposes. Our main result establishes that the Gaeta topos classifies exactly this theory of central-free models. Equivalently, the Gaeta topology encodes the central decompositions of finitely presented algebras and forces the generic central decomposition induced by its covering families to become trivial.
We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers ±1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results.
A 𝒞-set is a functor from a category 𝒞 to the category of finite sets, and 𝒞-set denotes the category of such functors with natural transformations as morphisms. In this work, we prove that 𝒞 is a groupoid if and only if 𝒞-set has finitely many indecomposable objects, which is the categorical analog of transitive G-sets in classical group actions.
We characterise the frame morphisms f:L→ M that lift to frame maps f:_b(L)→_b(M) , where _b(L) is the collection of joins of complemented sublocales of a frame L, or equivalently the Booleanization of the collection (L) of all its sublocales. We do so by proving that _b(L) is isomorphic to the Bruns–Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form 𝔠(a)∩𝔬(b) for a,b∈ L .
This paper introduces and investigates the category CL _𝒵 of 𝒵 -closure spaces. These spaces are defined based on a subset system 𝒵 on the category CLAT of complete lattices. Specifically, a closure space X is called a 𝒵 -closure space if its lattice of closed sets Γ (X) is closed under unions of 𝒵 -sets (members of 𝒵(Γ (X)) ). We introduce the concept of 𝒵 -irreducible sets and define an associated subset system 𝒵_I on the category CL _0 of T_0 closure spaces. Utilizing the theory of Z-completions and the b-closure operator, we provide a complete characterization of reflective subcategories of CL _𝒵 that contain a specific non- T_1 space (which is pointed under certain conditions). This characterization links reflectivity to the properties of being a K-category and being equivalent to a category CCS _Z of Z-convergence 𝒵 -closure spaces, where Z is a subset system coarser than 𝒵_I on CL _𝒵 . Furthermore, we present a unified construction for the reflective hull of subcategories within CL _𝒵 . Finally, we apply the main results to several concrete category instances, including closure spaces ( CL _0 ), topological spaces ( TOP _0 ), P-spaces ( PTOP _0 ), convex spaces ( CONV _0 ), and Alexandroff spaces ( ALEX _0 ), thereby unifying and generalizing results concerning reflectivity in these categories.
We describe a simple criterion which makes it easy to recognise when a pseudomonad is lax-idempotent. The criterion concerns the behaviour of colax bilimits of arrows - certain comma objects - and is easy to verify in examples. Building on this, we obtain a new characterisation of lax-idempotent pseudomonads on 2-categories with colax bilimits of arrows.
We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of coalgebraic models of algebraic theories endowed with an underlying structure of cocommutative Hopf algebra, and show that these categories are semi-abelian. We call them “categories of Ω -Hopf algebras”, since it is possible to characterize them as coalgebraic models of algebraic theories of Ω -groups in the sense of Higgins.
We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras, cluster varieties, cluster categories and surface models all have associated abstract cluster structures. For the first two classes, we also show that they can be constructed from abstract cluster structures. By defining a suitable notion of morphism of abstract cluster structures, we introduce a category of these and show that it has several desirable properties, such as initial and terminal objects and finite products and coproducts. We also prove that rooted cluster morphisms of cluster algebras give rise to morphisms of the associated abstract cluster structures, so that our framework includes a version of the extant category of cluster algebras. We can do more, however, because we can relate different types of representation of abstract cluster structures (cluster algebra, varieties, categories) directly via morphisms of their associated abstract cluster structures, even though no direct map from e.g. a cluster category to the associated cluster algebra is possible. In fact, we do much of the above in the setting of abstract quantum cluster structures, with some analysis of the difference between the category of these and that of the unquantized version. In order to show the relationship between abstract quantum cluster structures and quantum cluster algebras, we reformulate the usual construction of the latter in a way that is more amenable to our purposes and which we expect will be of independent interest and use.
We introduce a notion of (co)presheaf on a lax double functor X, which we generally call an instance. In the terminology of double-categorical logic, a lax double functor valued in sets, possibly preserving finite products, is called a model of a double (Lawvere) theory. By varying the double theory, we uniformly define a well-behaved notion of instances of categories, profunctors, monads, monoidal categories, multicategories, and more, and we recover for instance the multifunctors into the category of sets in the last example. We show that instances of X can be described either in terms of modules from the terminal model I to X, satisfying an additional condition on triviality of the left action, or as loose natural transformations from I to X. We propose a notion of discrete opfibration between models of a double theory, establish a comprehensive factorization system, and prove an elements correspondence giving an equivalence between the category of instances of and the category of discrete opfibrations over a model X. We describe properties of the resulting categories of instances, relying on a "collage" construction which we characterize as a lax colimit of a model of a double theory. An appendix gives a detailed treatment of certain morphisms of lax functors relevant also for bicategory theory: (loose) transformations versus modules and modifications versus modulations.
As already mentioned by Lawvere in his 1973 paper, the characterisation of Cauchy completeness of metric spaces in terms of representability of adjoint distributors amounts to the idempotent-split property of an ordinary category when the governing symmetric monoidal-closed category is changed from the extended real half-line to the category of sets. In this paper, for any commutative quantale 𝒱 taking the role of [0,∞ ] , we extend these two characterisations of Lawvere-style completeness from ordinary to 𝒱 -normed categories, that is, to categories enriched in the category of 𝒱 -normed sets, i.e., of sets equipped with a 𝒱 -valued function. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important 𝒱 -normed categories.
This is an overview of double categories of "open systems": systems that can interact with their environment. We focus on the variable sharing paradigm, where we compose open systems by identifying variables. This paradigm is often implemented using structured or decorated cospans. We explain this approach using three main examples: open Petri nets, open dynamical systems, and open Petri nets with rates. We compare the virtues of structured and decorated cospan double categories, and study their common features. We show that any symmetric monoidal structured or decorated cospan double category comes with maps from two simpler double categories: its "exoskeleton" and its "outer shell". Finally, we study the concept of "hypergraph double category", a kind of double category that should subsume structured and decorated cospans in a common framework for studying open systems in the variable sharing paradigm.
Frobenius algebras in the category of sets and relations ( Rel ) serve as a unifying framework for various algebraic and combinatorial structures, including groupoids, effect algebras, and abstract circles. Recently, a nerve construction of simplicial sets for Frobenius algebras in Rel has been introduced. In this work, we investigate the lifting properties of these simplicial sets, linking them to the algebraic properties of Frobenius algebras. We introduce ε -simplicial sets—simplicial sets with marked edges—that enable the representation of a broader class of structures, such as test spaces from quantum logic. Our main results focus on weakly saturated classes generated by cofibrations, corresponding to specific lifting problems. Furthermore, we provide a characterization of Frobenius algebras in Rel within the framework of ε -simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research.
