{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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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