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Soichiro Fujii

Publications and source records attributed to Soichiro Fujii.

At least 19 recordsLinked to original sources
arXiv2026

Monads and Distributive Laws in Substructural Contexts (Extended Version)

We present a categorical theory of monads and distributive laws in substructural contexts. In the study of distributive laws, the roles of (the absence of) structural rules for variable contexts have been recognized; our theory formalizes these substructural situations using Tronin's verbal categories $\mathbf W$, in a uniform and presentation-independent manner. We introduce the classes of $\mathbf W$-operadic monads (those defined via the structural rules in $\mathbf W$) and of $\mathbf W$-commutative monads (those invariant under the structural rules in $\mathbf W$). We give a canonical construction of a distributive law $ST\to TS$ of monads on $\mathbf{Set}$; it is applicable when $S$ is $\mathbf W$-operadic and $T$ is $\mathbf W$-commutative (under mild conditions). This accounts for many known and new distributive laws. Even when $S$ fails to be $\mathbf W$-operadic, we can refine $S$ and force $\mathbf W$-operadicity; this captures Varacca and Winskel's construction of indexed valuations.

cs.LO↗
arXiv2025

Nerves of generalized multicategories

For any category ${\mathcal E}$ and monad $T$ thereon, we introduce the notion of $T$-simplicial object in ${\mathcal E}$. Any $T$-category in the sense of Burroni induces a $T$-simplicial object as its nerve. This nerve construction defines a fully faithful functor from the category $\mathbf{Cat}_T({\mathcal E})$ of $T$-categories to the category $s_T({\mathcal E})$ of $T$-simplicial objects, whose essential image is characterized by a simple condition. We show that the category $s_T({\mathcal E})$ is enriched over the category of simplicial sets, and that this induces the usual 2-category structure on $\mathbf{Cat}_T({\mathcal E})$. We also study enriched limits and colimits in $s_T({\mathcal E})$ and $\mathbf{Cat}_T({\mathcal E})$, and show that if ${\mathcal E}$ is locally finitely presentable and $T$ is finitary, then $\mathbf{Cat}_T({\mathcal E})$ is locally finitely presentable as a 2-category and $s_T({\mathcal E})$ is locally finitely presentable as a simplicially-enriched category.

math.CT↗
arXiv2025

$ω$-equifibrations between strict and weak $ω$-categories

We study $ω$-equifibrations between weak $ω$-categories in the sense of Batanin--Leinster. We define $ω$-equifibrations as a natural weak $ω$-categorical analogue of isofibrations between categories, and show that they can be characterised via the right lifting property with respect to a suitable set $J$ of strict $ω$-functors. The definition of $J$ involves the construction of a certain weak $ω$-category $\mathcal{E}^1$ which, roughly speaking, is freely generated by an equivalence 1-cell in a ``coherent'' manner. We show that the strict version of $\mathcal{E}^1$ coincides with Ozornova and Rovelli's coherent walking $ω$-equivalence $\widehat{ω\mathcal{E}}$. The $ω$-equifibrations between strict $ω$-categories coincide with the fibrations in the folk model structure.

math.CT↗
arXiv2025

Homotopy types of Hom complexes of graph homomorphisms whose codomains are square-free

Given finite simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex having the graph homomorphisms $G\to H$ as the vertices. We determine the homotopy type of each connected component of $\mathrm{Hom}(G,H)$ when $H$ is square-free, meaning that it does not contain the $4$-cycle graph $C_4$ as a subgraph. Specifically, for a connected $G$ and a square-free $H$, we show that each connected component of $\mathrm{Hom}(G,H)$ is homotopy equivalent to a wedge sum of circles. We further show that, given any graph homomorphism $f\colon G\to H$ to a square-free $H$, one can determine the homotopy type of the connected component of $\mathrm{Hom}(G,H)$ containing $f$ algorithmically.

math.CO↗
arXiv2025

The familial nature of enrichment over virtual double categories

Originally enriched categories were defined over a monoidal category, but it was gradually realized that important examples can only be included when one enriches over more general structures such as bicategories and virtual double categories. We show that, as well as allowing more examples, working over virtual double categories also gives better formal properties. We study the 2-functor sending a virtual double category to the 2-category of categories enriched over it. We show that this is a parametric right 2-adjoint, and in fact is familial. We also show how a ``families construction'' for virtual double categories can be used to give a formal construction of the 2-category of categories enriched over a virtual double category.

math.CT↗
arXiv2025

Homotopy types of Hom complexes of graph homomorphisms whose codomains are cycles

For simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex whose vertices are the graph homomorphisms $G\to H$ and whose edges connect the pairs of homomorphisms which differ in a single vertex of $G$. Hom complexes play an important role in an algebro-topological approach to the graph coloring problem. It is known that $\mathrm{Hom}(G,H)$ is homotopy equivalent to a disjoint union of points and circles when both $G$ and $H$ are cycles. We generalize this known result by showing that the same holds whenever $G$ is connected and $H$ is a cycle. To this end, we explicitly construct the universal cover of each connected component of $\mathrm{Hom}(G,H)$ and prove that it is contractible. Additionally, we provide a simple criterion to determine whether the connected component containing a given homomorphism is homotopy equivalent to a point or circle.

math.CO↗
arXiv2025

$ω$-weak equivalences between weak $ω$-categories

We study $ω$-weak equivalences between weak $ω$-categories in the sense of Batanin-Leinster. Our $ω$-weak equivalences are strict $ω$-functors satisfying essential surjectivity in every dimension, and when restricted to those between strict $ω$-categories, they coincide with the weak equivalences in the model category of strict $ω$-categories defined by Lafont, Métayer, and Worytkiewicz. We show that the class of $ω$-weak equivalences has the 2-out-of-3 property. We also consider a generalisation of $ω$-weak equivalences, defined as weak $ω$-functors (in the sense of Garner) satisfying essential surjectivity, and show that this class also has the 2-out-of-3 property.

math.CT↗
arXiv2024

Weakly invertible cells in a weak $ω$-category

We study weakly invertible cells in weak $ω$-categories in the sense of Batanin-Leinster, adopting the coinductive definition of weak invertibility. We show that weakly invertible cells in a weak $ω$-category are closed under globular pasting. Using this, we generalise elementary properties of weakly invertible cells known to hold in strict $ω$-categories to weak $ω$-categories, and show that every weak $ω$-category has a largest weak $ω$-subgroupoid.

math.CT↗
arXiv2024

The oplax limit of an enriched category

We show that 2-categories of the form $\mathscr{B}\mbox{-}\mathbf{Cat}$ are closed under slicing, provided that we allow $\mathscr{B}$ to range over bicategories (rather than, say, monoidal categories). That is, for any $\mathscr{B}$-category $\mathbb{X}$, we define a bicategory $\mathscr{B}/\mathbb{X}$ such that $\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong (\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$. The bicategory $\mathscr{B}/\mathbb{X}$ is characterized as the oplax limit of $\mathbb{X}$, regarded as a lax functor from a chaotic category to $\mathscr{B}$, in the 2-category $\mathbf{BICAT}$ of bicategories, lax functors and icons. We prove this conceptually, through limit-preservation properties of the 2-functor $\mathbf{BICAT}\to 2\mbox{-}\mathbf{CAT}$ which maps each bicategory $\mathscr{B}$ to the 2-category $\mathscr{B}\mbox{-}\mathbf{Cat}$. When $\mathscr{B}$ satisfies a mild local completeness condition, we also show that the isomorphism $\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong (\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$ restricts to a correspondence between fibrations in $\mathscr{B}\mbox{-}\mathbf{Cat}$ over $\mathbb{X}$ on the one hand, and $\mathscr{B}/\mathbb{X}$-categories admitting certain powers on the other.

math.CT↗
arXiv2023

Ordered semirings and subadditive morphisms

An ordered semiring is a commutative semiring equipped with a compatible preorder. Ordered semirings generalise both distributive lattices and commutative rings, and provide a convenient framework to unify certain aspects of lattice theory and ring theory. The ideals of an ordered semiring $A$ form a commutative integral quantale $\mathrm{Idl}(A)$, and similarly, the radical ideals of $A$ form a (spatial) frame $\mathrm{Rad}(A)$. We characterise $\mathrm{Idl}$ and $\mathrm{Rad}$ as the left adjoints of the (non-full) inclusion functors from the categories of commutative integral quantales and of frames, respectively, to that of ordered semirings and subadditive morphisms between them. The (sober) topological space $\mathrm{pt}(\mathrm{Rad}(A))$ corresponding to $\mathrm{Rad}(A)$ is homeomorphic to the space $\mathrm{Spec}(A)$ of prime ideals of $A$.

math.CT↗
arXiv2022

Quantaloidal Approach to Constraint Satisfaction

The constraint satisfaction problem (CSP) is a computational problem that includes a range of important problems in computer science. We point out that fundamental concepts of the CSP, such as the solution set of an instance and polymorphisms, can be formulated abstractly inside the 2-category PFinSet of finite sets and sets of functions between them. The 2-category PFinSet is a quantaloid, and the formulation relies mainly on structure available in any quantaloid. This observation suggests a formal development of generalisations of the CSP and concomitant notions of polymorphism in a large class of quantaloids. We extract a class of optimisation problems as a special case, and show that their computational complexity can be classified by the associated notion of polymorphism.

math.CT↗
arXiv2021

Hom weak $ω$-categories of a weak $ω$-category

Classical definitions of weak higher-dimensional categories are given inductively; for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak $n$-categories for all natural numbers $n$, or of weak $ω$-categories, take more sophisticated approaches, and the nature of the "hom" is often not immediate from the definitions. In this paper, we focus on Leinster's definition of weak $ω$-category based on an earlier definition by Batanin, and construct for each weak $ω$-category $\mathcal{A}$, an underlying (weak $ω$-category)-enriched graph consisting of the same objects and for each pair of objects $x$ and $y$, a hom weak $ω$-category $\mathcal{A}(x,y)$. We also show that our construction is functorial with respect to weak $ω$-functors introduced by Garner.

math.CT↗
arXiv2021

Completeness and injectivity

We show that for any quantale $\mathcal{Q}$, a $\mathcal{Q}$-category is skeletal and complete if and only if it is injective with respect to fully faithful $\mathcal{Q}$-functors. This is a special case of known theorems due to Hofmann and Stubbe, but we provide a different proof, using the characterisation of the MacNeille completion of a $\mathcal{Q}$-category as its injective envelope. For Lawvere metric spaces, our results yield those of Kemajou, Künzi and Otafudu. We point out that their notion of Isbell convexity can be seen as a geometric formulation of categorical completeness for Lawvere metric spaces.

math.CT↗
arXiv2020

Enriched categories and tropical mathematics

This is a survey paper on the connection of enriched category theory over a quantale and tropical mathematics. Quantales or complete idempotent semirings, as well as matrices with coefficients in them, are fundamental objects in both fields. We first explain standard category-theoretic constructions on matrices, namely composition, right extension, right lifting and the Isbell hull. Along the way, we review known reformulations (due to Elliott and Willerton) of tropical polytopes, directed tight spans and the Legendre--Fenchel transform by means of these constructions, illustrating their ubiquity in tropical mathematics and related fields. We then consider complete semimodules over a quantale $\mathcal{Q}$, a tropical analogue of vector spaces over a field, and mention Stubbe's result identifying them with skeletal and complete $\mathcal{Q}$-categories. With the aim to bridge a gap between enriched category theory and tropical mathematics, we assume no knowledge in either field.

math.CT↗
arXiv2019

A unified framework for notions of algebraic theory

Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this paper, we develop a unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that algebraic theories correspond to monoid objects therein. To incorporate semantics, we introduce a categorical structure called metamodel, which formalises a definition of models of algebraic theories. We also define morphisms between notions of algebraic theory, which are a monoidal version of profunctors. Every strong monoidal functor gives rise to an adjoint pair of such morphisms, and provides a uniform method to establish isomorphisms between categories of models in different notions of algebraic theory. A general structure-semantics adjointness result and a double categorical universal property of categories of models are also shown.

math.CT↗
arXiv2019

A 2-Categorical Study of Graded and Indexed Monads

In the study of computational effects, it is important to consider the notion of computational effects with parameters. The need of such a notion arises when, for example, statically estimating the range of effects caused by a program, or studying the ways in which effects with local scopes are derived from effects with only the global scope. Extending the classical observation that computational effects can be modeled by monads, these computational effects with parameters are modeled by various mathematical structures including graded monads and indexed monads, which are two different generalizations of ordinary monads. The former has been employed in the semantics of effect systems, whereas the latter in the study of the relationship between the local state monads and the global state monads, each exemplifying the two situations mentioned above. However, despite their importance, the mathematical theory of graded and indexed monads is far less developed than that of ordinary monads. Here we develop the mathematical theory of graded and indexed monads from a 2-categorical viewpoint. We first introduce four 2-categories and observe that in two of them graded monads are in fact monads in the 2-categorical sense, and similarly indexed monads are monads in the 2-categorical sense in the other two. We then construct explicitly the Eilenberg--Moore and the Kleisli objects of graded monads, and the Eilenberg--Moore objects of indexed monads in the sense of Street in appropriate 2-categories among these four. The corresponding results for graded and indexed comonads also follow. We expect that the current work will provide a theoretical foundation to a unified study of computational effects with parameters, or dually (using the comonad variants), of computational resources with parameters, arising for example in Bounded Linear Logic.

math.CT↗
arXiv2019

A Categorical Approach to L-Convexity

We investigate an enriched-categorical approach to a field of discrete mathematics. The main result is a duality theorem between a class of enriched categories (called $\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-categories) and that of what we call ($\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-) extended L-convex sets. We introduce extended L-convex sets as variants of certain discrete structures called L-convex sets and L-convex polyhedra, studied in the field of discrete convex analysis. We also introduce homomorphisms between extended L-convex sets. The theorem claims that there is a one to one correspondence (up to isomorphism) between two classes. The thesis also contains an introductory chapter on enriched categories and no categorical knowledge is assumed.

math.CT↗