We study coinductive invertibility of cells in weak $omega$-categories. We use the inductive presentation of weak $omega$-categories via an adjunction with the category of computads, and show that invertible cells are closed under all operations of $omega$-categories. Moreover, we give a simple criterion for invertibility in computads, together with an algorithm computing the data witnessing the invertibility, including the inverse, and the cancellation data.
{"title":"Invertible cells in $ω$-categories","authors":"Thibaut Benjamin, Ioannis Markakis","doi":"arxiv-2406.12127","DOIUrl":"https://doi.org/arxiv-2406.12127","url":null,"abstract":"We study coinductive invertibility of cells in weak $omega$-categories. We\u0000use the inductive presentation of weak $omega$-categories via an adjunction\u0000with the category of computads, and show that invertible cells are closed under\u0000all operations of $omega$-categories. Moreover, we give a simple criterion for\u0000invertibility in computads, together with an algorithm computing the data\u0000witnessing the invertibility, including the inverse, and the cancellation data.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this work is to further develop the calculus of (internal) relations for a regular Ord-category C. To capture the enriched features of a regular Ord-category and obtain a good calculus, the relations we work with are precisely the ideals in C. We then focus on an enriched version of the 1-dimensional algebraic 2-permutable (also called Mal'tsev) property and its well-known equivalent characterisations expressed through properties on ordinary relations. We introduce the notion of Ord-Mal'tsev category and show that these may be characterised through enriched versions of the above mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional Mal'tsev category is necessarily an Ord-Mal'tsev category. We also give some examples of categories which are not Mal'tsev categories, but are Ord-Mal'tsev categories.
为了捕捉正则表达式范畴 C 的丰富特征并获得良好的微积分，我们所处理的关系正是 C 中的理想关系。然后，我们聚焦于一维代数 2-可变（也称为 Mal'tsev）性质的丰富版本，以及通过关于普通关系的性质所表达的其众所周知的等价特征。我们引入了 Ord-Mal'tsev 范畴的概念，并证明它们可以通过上述性质的丰富版本来表征理想。一维马氏范畴的任何 Ord-enrichment 都必然是一个 Ord-Mal'tsev 范畴。我们还给出了一些不是Mal'tsev范畴，但却是Ord-Mal'tsev范畴的例子。
{"title":"Enriched aspects of calculus of relations and $2$-permutability","authors":"Maria Manuel Clementino, Diana Rodelo","doi":"arxiv-2406.10624","DOIUrl":"https://doi.org/arxiv-2406.10624","url":null,"abstract":"The aim of this work is to further develop the calculus of (internal)\u0000relations for a regular Ord-category C. To capture the enriched features of a\u0000regular Ord-category and obtain a good calculus, the relations we work with are\u0000precisely the ideals in C. We then focus on an enriched version of the\u00001-dimensional algebraic 2-permutable (also called Mal'tsev) property and its\u0000well-known equivalent characterisations expressed through properties on\u0000ordinary relations. We introduce the notion of Ord-Mal'tsev category and show\u0000that these may be characterised through enriched versions of the above\u0000mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional\u0000Mal'tsev category is necessarily an Ord-Mal'tsev category. We also give some\u0000examples of categories which are not Mal'tsev categories, but are Ord-Mal'tsev\u0000categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the categorical properties of right-preordered groups, giving an explicit description of limits and colimits in this category, and studying some exactness properties. We show that, from an algebraic point of view, the category of right-preordered groups shares several properties with the one of monoids. Moreover, we describe split extensions of right-preordered groups, showing in particular that semidirect products of ordered groups have always a natural right-preorder.
{"title":"Right-preordered groups from a categorical perspective","authors":"Maria Manuel Clementino, Andrea Montoli","doi":"arxiv-2406.10071","DOIUrl":"https://doi.org/arxiv-2406.10071","url":null,"abstract":"We study the categorical properties of right-preordered groups, giving an\u0000explicit description of limits and colimits in this category, and studying some\u0000exactness properties. We show that, from an algebraic point of view, the\u0000category of right-preordered groups shares several properties with the one of\u0000monoids. Moreover, we describe split extensions of right-preordered groups,\u0000showing in particular that semidirect products of ordered groups have always a\u0000natural right-preorder.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend Lurie's definition of enriched $infty$-categories to notions of left enriched, right enriched and bi-enriched $infty$-categories, which generalize the concepts of closed left tensored, right tensored and bitensored $infty$-categories and share many desirable features with them. We use bi-enriched $infty$-categories to endow the $infty$-category of enriched functors with enrichment that generalizes both the internal hom of the tensor product of enriched $infty$-categories when the latter exists, and the free cocompletion under colimits and tensors. As an application we prove an end formula for morphism objects of enriched $infty$-categories of enriched functors and compute the monad for enriched functors. We build our theory closely related to Lurie's higher algebra: we construct an enriched $infty$-category of enriched presheaves via the enveloping tensored $infty$-category, construct transfer of enrichment via scalar extension of bitensored $infty$-categories, and construct enriched Kan-extensions via operadic Kan extensions. In particular, we develop an independent theory of enriched $infty$-categories for Lurie's model of enriched $infty$-categories.
{"title":"On bi-enriched $infty$-categories","authors":"Hadrian Heine","doi":"arxiv-2406.09832","DOIUrl":"https://doi.org/arxiv-2406.09832","url":null,"abstract":"We extend Lurie's definition of enriched $infty$-categories to notions of\u0000left enriched, right enriched and bi-enriched $infty$-categories, which\u0000generalize the concepts of closed left tensored, right tensored and bitensored\u0000$infty$-categories and share many desirable features with them. We use\u0000bi-enriched $infty$-categories to endow the $infty$-category of enriched\u0000functors with enrichment that generalizes both the internal hom of the tensor\u0000product of enriched $infty$-categories when the latter exists, and the free\u0000cocompletion under colimits and tensors. As an application we prove an end\u0000formula for morphism objects of enriched $infty$-categories of enriched\u0000functors and compute the monad for enriched functors. We build our theory\u0000closely related to Lurie's higher algebra: we construct an enriched\u0000$infty$-category of enriched presheaves via the enveloping tensored\u0000$infty$-category, construct transfer of enrichment via scalar extension of\u0000bitensored $infty$-categories, and construct enriched Kan-extensions via\u0000operadic Kan extensions. In particular, we develop an independent theory of\u0000enriched $infty$-categories for Lurie's model of enriched $infty$-categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"153 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a theory of weighted colimits in the framework of weakly bi-enriched $infty$-categories, an extension of Lurie's notion of enriched $infty$-categories. We prove an existence result for weighted colimits, study weighted colimits of diagrams of enriched functors, express weighted colimits via enriched coends, characterize the enriched $infty$-category of enriched presheaves as the free cocompletion under weighted colimits and develop a theory of universally adjoining weighted colimits to an enriched $infty$-category. We use the latter technique to construct for every presentably $mathbb{E}_{k+1}$-monoidal $infty$-category $mathcal{V}$ for $1 leq k leq infty$ and class $mathcal{H}$ of $mathcal{V}$-weights, with respect to which weighted colimits are defined, a presentably $mathbb{E}_k$-monoidal structure on the $infty$-category of $mathcal{V}$-enriched $infty$-categories that admit $mathcal{H}$-weighted colimits. Varying $mathcal{H}$ this $mathbb{E}_k$-monoidal structure interpolates between the tensor product for $mathcal{V}$-enriched $infty$-categories and the relative tensor product for $infty$-categories presentably left tensored over $mathcal{V}$. As an application we prove that forming $mathcal{V}$-enriched presheaves is $mathbb{E}_k$-monoidal, construct a $mathcal{V}$-enriched version of Day-convolution and give a new construction of the tensor product for $infty$-categories presentably left tensored over $mathcal{V}$ as a $mathcal{V}$-enriched localization of Day-convolution. As further applications we construct a tensor product for Cauchy-complete $mathcal{V}$-enriched $infty$-categories, a tensor product for $(infty,2)$-categories with (op)lax colimits and a tensor product for stable $(infty,n)$-categories.
我们在弱偏富集$infty$类的框架内发展了加权冒点理论，这是对卢里的富集$infty$类概念的扩展。我们证明了加权余弦的存在性结果，研究了富集函子图的加权余弦，通过富集余弦表达了加权余弦，描述了富集预波的富集$infty$-类作为加权余弦下的自由共包的特征，并发展了普遍邻接富集$infty$-类的加权余弦的理论。我们使用后一种技术来为1leq k leqinfty$和$mathcal{V}$-weights的类$mathcal{H}$构建每个现存的$mathbb{E}_{k+1}$-单元$infty$-类$mathcal{V}$、相对于定义了加权 colimits 的 $mathcal{V}$ 类，在允许 $mathcal{H}$ 加权 colimits 的 $infty$ 类上有一个现成的 $mathbb{E}_k$ 单元结构。随着$mathcal{H}$的变化，这个$mathbb{E}_k$单元结构会在为mathcal{V}$富集的$infty$范畴的张量积与为infty$范畴在$mathcal{V}$上呈现出的左张量的相对张量积之间进行调节。作为一个应用，我们证明了形成$mathcal{V}$-enriched presheaves是$mathbb{E}_k$-monoidal的，构造了一个$mathcal{V}$-enriched版本的Day-convolution，并给出了一个新的构造，即作为一个$mathcal{V}$-enriched localization of Day-convolution的$infty$-categories presentably left tensored over$mathcal{V}$ 的张量积。作为进一步的应用，我们为Cauchy-complete$mathcal{V}$-enriched $infty$-categories构造了一个张量积，为具有(op)lax colimits的$(infty,2)$-categories构造了一个张量积，为稳定的$(infty,n)$-categories构造了一个张量积。
{"title":"The higher algebra of weighted colimits","authors":"Hadrian Heine","doi":"arxiv-2406.08925","DOIUrl":"https://doi.org/arxiv-2406.08925","url":null,"abstract":"We develop a theory of weighted colimits in the framework of weakly\u0000bi-enriched $infty$-categories, an extension of Lurie's notion of enriched\u0000$infty$-categories. We prove an existence result for weighted colimits, study\u0000weighted colimits of diagrams of enriched functors, express weighted colimits\u0000via enriched coends, characterize the enriched $infty$-category of enriched\u0000presheaves as the free cocompletion under weighted colimits and develop a\u0000theory of universally adjoining weighted colimits to an enriched\u0000$infty$-category. We use the latter technique to construct for every\u0000presentably $mathbb{E}_{k+1}$-monoidal $infty$-category $mathcal{V}$ for $1\u0000leq k leq infty$ and class $mathcal{H}$ of $mathcal{V}$-weights, with\u0000respect to which weighted colimits are defined, a presentably\u0000$mathbb{E}_k$-monoidal structure on the $infty$-category of\u0000$mathcal{V}$-enriched $infty$-categories that admit $mathcal{H}$-weighted\u0000colimits. Varying $mathcal{H}$ this $mathbb{E}_k$-monoidal structure\u0000interpolates between the tensor product for $mathcal{V}$-enriched\u0000$infty$-categories and the relative tensor product for $infty$-categories\u0000presentably left tensored over $mathcal{V}$. As an application we prove that\u0000forming $mathcal{V}$-enriched presheaves is $mathbb{E}_k$-monoidal, construct\u0000a $mathcal{V}$-enriched version of Day-convolution and give a new construction\u0000of the tensor product for $infty$-categories presentably left tensored over\u0000$mathcal{V}$ as a $mathcal{V}$-enriched localization of Day-convolution. As\u0000further applications we construct a tensor product for Cauchy-complete\u0000$mathcal{V}$-enriched $infty$-categories, a tensor product for\u0000$(infty,2)$-categories with (op)lax colimits and a tensor product for stable\u0000$(infty,n)$-categories.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A quick overview of category theory and topos theory including slice categories, monics, epics, isos, diagrams, cones, cocones, limits, colimits, products and coproducts, pushouts and pullbacks, equalizers and coequalizers, initial and terminal objects, exponential objects, subobjects, subobject classifiers, the definition of a topos, algebras of subobjects, functors, natural transformations and adjoint functors. This paper is refashioned and adopted from Richard Pettigrew's university notes.
{"title":"A Very Short Introduction to Topos Theory (adapted from Prof. Pettigrew's notes)","authors":"Eric Schmid","doi":"arxiv-2406.19409","DOIUrl":"https://doi.org/arxiv-2406.19409","url":null,"abstract":"A quick overview of category theory and topos theory including slice\u0000categories, monics, epics, isos, diagrams, cones, cocones, limits, colimits,\u0000products and coproducts, pushouts and pullbacks, equalizers and coequalizers,\u0000initial and terminal objects, exponential objects, subobjects, subobject\u0000classifiers, the definition of a topos, algebras of subobjects, functors,\u0000natural transformations and adjoint functors. This paper is refashioned and adopted from Richard Pettigrew's university\u0000notes.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"167 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
When a category is equipped with a 2-cell structure it becomes a sesquicategory but not necessarily a 2-category. It is widely accepted that the latter property is equivalent to the middle interchange law. However, little attention has been given to the study of the category of all 2-cell structures (seen as sesquicategories with a fixed underlying base category) other than as a generalization for 2-categories. The purpose of this work is to highlight the significance of such a study, which can prove valuable in identifying intrinsic features pertaining to the base category. These ideas are expanded upon through the guiding example of the category of monoids. Specifically, when a monoid is viewed as a one-object category, its 2-cell structures resemble semibimodules.
{"title":"On categories with arbitrary 2-cell structures","authors":"Nelson Martins-Ferreira","doi":"arxiv-2406.08240","DOIUrl":"https://doi.org/arxiv-2406.08240","url":null,"abstract":"When a category is equipped with a 2-cell structure it becomes a\u0000sesquicategory but not necessarily a 2-category. It is widely accepted that the\u0000latter property is equivalent to the middle interchange law. However, little\u0000attention has been given to the study of the category of all 2-cell structures\u0000(seen as sesquicategories with a fixed underlying base category) other than as\u0000a generalization for 2-categories. The purpose of this work is to highlight the\u0000significance of such a study, which can prove valuable in identifying intrinsic\u0000features pertaining to the base category. These ideas are expanded upon through\u0000the guiding example of the category of monoids. Specifically, when a monoid is\u0000viewed as a one-object category, its 2-cell structures resemble semibimodules.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This text is dedicated to the development of the theory of $(infty,omega)$-categories. We present generalizations of standard results from category theory, such as the lax Grothendieck construction, the Yoneda lemma, lax (co)limits and lax Kan extensions, among others.
{"title":"Categorical Theory of $(infty,ω)$-Categories","authors":"Félix Loubaton","doi":"arxiv-2406.05425","DOIUrl":"https://doi.org/arxiv-2406.05425","url":null,"abstract":"This text is dedicated to the development of the theory of\u0000$(infty,omega)$-categories. We present generalizations of standard results\u0000from category theory, such as the lax Grothendieck construction, the Yoneda\u0000lemma, lax (co)limits and lax Kan extensions, among others.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce structured active inference, a large generalization and formalization of active inference using the tools of categorical systems theory. We cast generative models formally as systems "on an interface", with the latter being a compositional abstraction of the usual notion of Markov blanket; agents are then 'controllers' for their generative models, formally dual to them. This opens the active inference landscape to new horizons, such as: agents with structured interfaces (e.g. with 'mode-dependence', or that interact with computer APIs); agents that can manage other agents; and 'meta-agents', that use active inference to change their (internal or external) structure. With structured interfaces, we also gain structured ('typed') policies, which are amenable to formal verification, an important step towards safe artificial agents. Moreover, we can make use of categorical logic to describe express agents' goals as formal predicates, whose satisfaction may be dependent on the interaction context. This points towards powerful compositional tools to constrain and control self-organizing ensembles of agents.
{"title":"Structured Active Inference (Extended Abstract)","authors":"Toby St Clere Smithe","doi":"arxiv-2406.07577","DOIUrl":"https://doi.org/arxiv-2406.07577","url":null,"abstract":"We introduce structured active inference, a large generalization and\u0000formalization of active inference using the tools of categorical systems\u0000theory. We cast generative models formally as systems \"on an interface\", with\u0000the latter being a compositional abstraction of the usual notion of Markov\u0000blanket; agents are then 'controllers' for their generative models, formally\u0000dual to them. This opens the active inference landscape to new horizons, such\u0000as: agents with structured interfaces (e.g. with 'mode-dependence', or that\u0000interact with computer APIs); agents that can manage other agents; and\u0000'meta-agents', that use active inference to change their (internal or external)\u0000structure. With structured interfaces, we also gain structured ('typed')\u0000policies, which are amenable to formal verification, an important step towards\u0000safe artificial agents. Moreover, we can make use of categorical logic to\u0000describe express agents' goals as formal predicates, whose satisfaction may be\u0000dependent on the interaction context. This points towards powerful\u0000compositional tools to constrain and control self-organizing ensembles of\u0000agents.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This thesis revolves around an area of computer science called "semantics". We work with operational semantics, equational theories, and denotational semantics. The first contribution of this thesis is a study of the commutativity of effects through the prism of monads. Monads are the generalisation of algebraic structures such as monoids, which have a notion of centre: the centre of a monoid is made of elements which commute with all others. We provide the necessary and sufficient conditions for a monad to have a centre. We also detail the semantics of a language with effects that carry information on which effects are central. Moreover, we provide a strong link between its equational theories and its denotational semantics. The second focus of the thesis is quantum computing, seen as a reversible effect. Physically permissible quantum operations are all reversible, except measurement; however, measurement can be deferred at the end of the computation, allowing us to focus on the reversible part first. We define a simply-typed reversible programming language performing quantum operations called "unitaries". A denotational semantics and an equational theory adapted to the language are presented, and we prove that the former is complete. Furthermore, we study recursion in reversible programming, providing adequate operational and denotational semantics to a Turing-complete, reversible, functional programming language. The denotational semantics uses the dcpo enrichment of rig join inverse categories. This mathematical account of higher-order reasoning on reversible computing does not directly generalise to its quantum counterpart. In the conclusion, we detail the limitations and possible future for higher-order quantum control through guarded recursion.
{"title":"The Semantics of Effects: Centrality, Quantum Control and Reversible Recursion","authors":"Louis Lemonnier","doi":"arxiv-2406.07216","DOIUrl":"https://doi.org/arxiv-2406.07216","url":null,"abstract":"This thesis revolves around an area of computer science called \"semantics\".\u0000We work with operational semantics, equational theories, and denotational\u0000semantics. The first contribution of this thesis is a study of the commutativity of\u0000effects through the prism of monads. Monads are the generalisation of algebraic\u0000structures such as monoids, which have a notion of centre: the centre of a\u0000monoid is made of elements which commute with all others. We provide the\u0000necessary and sufficient conditions for a monad to have a centre. We also\u0000detail the semantics of a language with effects that carry information on which\u0000effects are central. Moreover, we provide a strong link between its equational\u0000theories and its denotational semantics. The second focus of the thesis is quantum computing, seen as a reversible\u0000effect. Physically permissible quantum operations are all reversible, except\u0000measurement; however, measurement can be deferred at the end of the\u0000computation, allowing us to focus on the reversible part first. We define a\u0000simply-typed reversible programming language performing quantum operations\u0000called \"unitaries\". A denotational semantics and an equational theory adapted\u0000to the language are presented, and we prove that the former is complete. Furthermore, we study recursion in reversible programming, providing adequate\u0000operational and denotational semantics to a Turing-complete, reversible,\u0000functional programming language. The denotational semantics uses the dcpo\u0000enrichment of rig join inverse categories. This mathematical account of\u0000higher-order reasoning on reversible computing does not directly generalise to\u0000its quantum counterpart. In the conclusion, we detail the limitations and\u0000possible future for higher-order quantum control through guarded recursion.","PeriodicalId":501135,"journal":{"name":"arXiv - MATH - Category Theory","volume":"2016 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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