JIří Rosický
We develop universal algebra over an enriched category $\mathcal K$ and relate it to finitary enriched monads over $\mathcal K$. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.
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Strongly finitary metric monads are too strong
该研究指出强有限度量单子过于强大，通过反例证明自由代数单子并非全部强有限，并揭示了1-基本单子的特性。
Finitary monads on $\mathsf{Pos}$ are characterized as the precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analagously, finitary enriched monads on $\mathsf{Pos}$ are characterized: here we work with varieties of coherent algebras which means that their operations are monotone.
Sifted Colimits, Strongly Finitary Monads and Continuous Algebras
本文刻画了 Pos、CPO 和 DCPO 范畴上强有限单子等价于保持筛选余极限的单子，并将连续代数簇等同于强有限单子的单子范畴。
We consider the equivalence of Lawvere theories and finitary monads on Set from the perspective of Endf(Set)-enriched category theory, where Endf(Set) is the category of finitary endofunctors of Set. We identify finitary monads with one-object Endf(Set)-categories, and ordinary categories admitting finite powers (i.e., n-fold products of each object with itself) with Endf(Set)-categories admitting a certain class Phi of absolute colimits; we then show that, from this perspective, the passage from a finitary monad to the associated Lawvere theory is given by completion under Phi-colimits. We al…
It is well known that classical varieties of $Σ$-algebras correspond bijectively to finitary monads on $\mathsf{Set}$. We present an analogous result for varieties of ordered $Σ$-algebras, i.e., classes presented by inequations between $Σ$-terms. We prove that they correspond bijectively to strongly finitary monads on $\mathsf{Pos}$. That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequaliser presentation of finitary monads due to Kelly and Power. We also show that these monads are lif…
For finitary regular monads T on locally finitely presentable categories we characterize the finitely presentable objects in the category of T-algebras in the style known from general algebra: they are precisely the algebras presentable by finitely many generators and finitely many relations.
An enriched view on the extended finitary monad--Lawvere theory correspondence
本文用丰富范畴论重新阐释有限余单子与Lawvere理论间的对应，将其视为丰富范畴在绝对余极限下的自由完备化，并创新性地使用双范畴丰富化以统一处理所有局部有限可表现基。
We develop a general framework for studying signatures, presentations, and algebraic colimits of enriched monads for a subcategory of arities, even when the base of enrichment $\mathcal{V}$ is not locally presentable. When $\mathcal{V}$ satisfies the weaker requirement of local boundedness, the resulting framework is sufficiently general to apply to the $Φ$-accessible monads of Lack and Rosický and the $\mathcal{J}$-ary monads of the first author, while even without local boundedness our framework captures in full generality the presentations of strongly finitary monads of Lack and Kelly as we…
We introduce a framework for universal algebra in categories of relational structures given by finitary relational signatures and finitary or infinitary Horn theories, with the arity $λ$ of a Horn theory understood as a strict upper bound on the number of premisses in its axioms; key examples include partial orders ($λ=ω$) or metric spaces ($λ=ω_1$). We establish a bijective correspondence between $λ$-accessible enriched monads on the given category of relational structures and a notion of $λ$-ary algebraic theories (i.e. with operations of arity $<λ$), with the syntax of algebraic theories in…
Discrete Lawvere theories and monads
本文研究了离散富集Lawvere理论与强有限富集单子之间的关系，证明了在特定条件下，强有限富集单子由离散富集Lawvere理论给出，并指出此类单子保持满射。
We define and study a notion of $\textit{commutant}$ for $\mathcal{V}$-enriched $\mathcal{J}$-algebraic theories for a system of arities $\mathcal{J}$, recovering the usual notion of commutant or centralizer of a subring as a special case alongside Wraith's notion of commutant for Lawvere theories as well as a notion of commutant for $\mathcal{V}$-monads on a symmetric monoidal closed category $\mathcal{V}$. This entails a thorough study of commutation and Kronecker products of operations in $\mathcal{J}$-theories. In view of the equivalence between $\mathcal{J}$-theories and $\mathcal{J}$-ary…
Quantitative Algebras and a Classification of Metric Monads
本文研究了定量代数与度量空间中单子的分类，证明了定量变种与强有限单子之间的双射关系，并扩展到一般富集单子与广义变种的对应。
We introduce the notion of a lax monoidal fibration and we show how it can be conveniently used to deal with various algebraic structures that play an important role in some definitions of the opetopic sets (Baez-Dolan, Hermida-Makkai-Power). We present the 'standard' such structures, the exponential fibrations of basic fibrations and three areas of applications. First area is related to the T-categories of A. Burroni. The monoids in the Burroni lax monoidal fibrations form the fibration of T-categories. The construction of the relative Burroni fibrations and free T-categories in this context,…
Strongly finitary monads and multi-sorted varieties enriched in cartesian closed concrete categories
本研究将富集范畴中代数簇与强有限单子的对偶等价结果，推广到任意具体笛卡尔闭范畴及多类情形，证明了对应范畴的对偶等价性，并给出了相关实例。
After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic theories to a general correspondence between monads and theories for a given category with arities. As application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids.
Birkhoff's variety theorem for relative algebraic theories
本文将集合范畴上代数理论与有限单子的等价性，通过部分霍恩逻辑推广到局部有限可表示范畴，定义$\boldsymbol{A}$-相对代数理论并证明其等价于该范畴上的有限单子，广义伯克霍夫多样性定理在此等价性证明中起关键作用。
Quantitative algebras are algebras enriched in the category $\mathsf{Met}$ of metric spaces so that all operations are nonexpanding. Mardare, Plotkin and Panangaden introduced varieties (aka $1$-basic varieties) as classes of quantitative algebras presented by quantitative equations. We prove that they bijectively correspond to strongly finitary monads $T$ on $\mathsf{Met}$. This means that $T$ is the Kan extension of its restriction to finite discrete spaces. An analogous result holds in the category $\mathsf{CMet}$ of complete metric spaces. Analogously, continuous algebras are algebras enri…
Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of universal algebra, can be generalized in three ways: replacing Set by another category, working in an enriched setting, and by working with another class of limits than finite products. An important special case involves working with sifted-colimit-preserving monads rather than filtered-colimit-preserving ones.
We investigate the foundations of a theory of algebraic data types with variable binding inside classical universal algebra. In the first part, a category-theoretic study of monads over the nominal sets of Gabbay and Pitts leads us to introduce new notions of finitary based monads and uniform monads. In a second part we spell out these notions in the language of universal algebra, show how to recover the logics of Gabbay-Mathijssen and Clouston-Pitts, and apply classical results from universal algebra.
The bounded ideal monad on the category of quasi-metric spaces and its algebras
本研究为拟度量空间引入有界理想概念，构造了拟度量空间范畴上的有界理想单子，刻画了其代数与连续代数的结构，建立了度量结构与序结构的联系。
Relativized universal algebra via partial Horn logic
本文基于部分Horn理论，将传统代数理论等价关系推广至任意局部可呈现范畴，提出相对代数理论概念，并推广Birkhoff定理构建相对普遍代数框架。
We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to be strictifiable. Our counterexample comes from higher category theory: the strict algebras are strict 3-categories, and the pseudoalgebras are a type of semi-strict 3-category lying in between Gray-categories and tricategories. Thus, the result follows from the fact that not every Gray-category is equivalent to a strict 3-category, connecting 2-categorical a…
Enriched structure-semantics adjunctions and monad-theory equivalences for subcategories of arities
本文为元数子范畴的充实结构-语义伴随与单子-理论等价发展了公理化框架，同时推广了既有理论并确立了适用于局部有界闭范畴的新示例类。
We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories enriched over a normal duoidal category; using this, we re-find notions such as the commutativity of a finitary algebraic theory or a strong monad, the commuting tensor product of two theories, and the Boardman-Vogt tensor product of symmetric operads.
For every finitary monad $T$ on sets and every endofunctor $F$ on the category of $T$-algebras we introduce the concept of an ffg-Elgot algebra for $F$, that is, an algebra admitting coherent solutions for finite systems of recursive equations with effects represented by the monad $T$. The goal is to study the existence and construction of free ffg-Elgot algebras. To this end, we investigate the locally ffg fixed point $φF$, i.e. the colimit of all $F$-coalgebras with free finitely generated carrier, which is shown to be the initial ffg-Elgot algebra. This is the technical foundation for our m…
The algebraic expression $3 + 2 + 6$ can be evaluated to $11$, but it can also be partially evaluated to $5 + 6$. In categorical algebra, such partial evaluations can be defined in terms of the $1$-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on $\mathsf{Set}$ (which are tho…
The quantum instrument monad
提出量子仪器单子 $\mathcal{I}_\mathcal{A}$，作为状态单子的非交换推广，用于建模与量子系统交互的计算效应，并给出集合范畴上的有限版本和可测空间范畴上的测度论版本。
We will show that Banach spaces are monadic over complete metric spaces via the unit ball functor. For the forgetful functor, one should take complete pointed metric spaces.
For a small quantaloid $\mathcal{Q}$, we consider 2-monads on the 2-category $\mathcal{Q}$-$\bf{Cat}$ and their lax extensions to the 2-category $\mathcal{Q}$-$\bf{Dist}$ of small $\mathcal{Q}$-categories and their distributors, in particular those lax extensions that are flat, in the sense that they map identity distributors to identity distributors. In fact, unlike in the discrete case, a 2-monad on $\mathcal{Q}$-$\bf{Cat}$ may admit only one flat lax extension. Every ordinary monad on the comma category $\bf{Set}/{\rm ob}\mathcal{Q}$ with a lax extension to $\mathcal{Q}$-$\bf{Rel}$ gives ri…
Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad
本文在连续quantale值度量空间框架中研究不精确性与鲁棒性，定义Hausdorff-Smyth单子并建立其诱导拓扑与幂集上鲁棒拓扑之间的关系。
For a quantaloid $\mathcal{Q}$, considered as a bicategory, Walters introduced categories enriched in $\mathcal{Q}$. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad $\mathbb{T}$ over the discrete presheaf monad of the small quantaloid $\mathcal{Q}$ the primary data of the theory, rather than the lax monad extensions of $\mathbb{T}$ to the category of $\mathcal{Q}$-relations that they equivalently describe. The central piece of the paper establishes a…
In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order. It can be considered a metric analogue of the probabilistic powerdomain. The spaces we consider, which we call L-ordered, are spaces where the order satisfies a mild compatibility condition with the metric itself, rather than merely with the underlying topology. As we show, this is relate…
Towards enriched universal algebra
本文提出了一种基于富集基础的普遍代数框架，通过定义语言和项的递归构造，将有限单调器的代数模型与等式理论联系起来。
It has been known since the work of Duskin and Pelletier four decades ago that KH^op, the category opposite to compact Hausdorff spaces and continuous maps, is monadic over the category of sets. It follows that KH^op is equivalent to a possibly infinitary variety of algebras V in the sense of Slominski and Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of V can be generated using a finite number of finitary operations, together with a single operation of countably infinite arity. In 1983, Banaschewski and Rosicky independently proved a conjecture of Bankston, establishi…
We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces. We prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limit…
We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec N) whose semiring of functions is (a P-version of) the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to Z and collapsing H_0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.
Nerves of generalized multicategories
本文研究了广义多范畴的神经构造及其在范畴理论中的应用，展示了T- simplicial对象的性质以及相关范畴的局部有限可现性。
We prove that the forgetful functors from the categories of $C^*$- and $W^*$-algebras to Banach $*$-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosický, and that the categories of unital (commutative) $C^*$-algebras are not locally-isometry $\aleph_0$-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosický. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are…
We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimono…
Barr-coexactness for metric compact Hausdorff spaces
本文在分离度量紧Hausdorff空间范畴中刻画正则单态射与满态射，给出用核度量表示商对象的方法，并证明其对偶范畴是Barr-正合的，推进了度量结构的范畴论研究。
Terminal Coalgebras in Countably Many Steps
该研究证明了局部有限呈现范畴上有限自函子的终余代数可通过终余代数链的可数极限得到，并给出Vietoris多项式函子等多类函子终余代数的步长结果及相关证明。
For single-sorted algebras, Fujiwara defined, through the concept of family of basic mapping-formulas, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras and an equivalence relation, the conjugation, on the families of basic mapping-formulas, which corresponds to the relation of inner isomorphism for algebras. In this paper we extend the theory of Fujiwara about morphisms to the many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures, which assigns to basic sorts, words and to formal operations, families of derived te…
Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:0804.0326 [math.CV]] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the category of injective spaces and left adjoints over $\mathsf{Set}$. In this paper we generalise these results, studying cocompleteness with respect to a given class of distributors. We show …
Our work is a fundamental study of the notion of approximation in V-categories and in (U,V)-categories, for a quantale V and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of V- and (U,V)-categories. We fully characterize continuous V-categories (resp. (U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in the same ways as continuous domains are characterized among all dcpos. By varying the choice of the quantale V and the notion of ideals, and by further allowing the ultrafilt…
Free algebras of topologically enriched multi-sorted equational theories
本文将多类等式理论推广到在 Set 上拓扑的对称幺半范畴 V 中强化的情形，证明自由 T-代数可由底层经典理论的自由代数提升得到，并给出显式归纳描述。
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