We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system), there is a canonical grading of T. Our application is to graded monads and models of computational effects. We demonstrate our results by characterizing the canonical gradings of a number of monads, for which C is endofunctors with composition. We also show that we can obtain canonical grades for algebraic operations.
{"title":"Canonical Gradings of Monads","authors":"Flavien Breuvart, Dylan McDermott, Tarmo Uustalu","doi":"10.4204/EPTCS.380.1","DOIUrl":"https://doi.org/10.4204/EPTCS.380.1","url":null,"abstract":"We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system), there is a canonical grading of T. Our application is to graded monads and models of computational effects. We demonstrate our results by characterizing the canonical gradings of a number of monads, for which C is endofunctors with composition. We also show that we can obtain canonical grades for algebraic operations.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"52 1","pages":"1-21"},"PeriodicalIF":0.0,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80985183","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}
Pub Date : 2023-01-01DOI: 10.48550/arXiv.2307.15519
{"title":"Proceedings Fifth International Conference on Applied Category Theory, ACT 2022, Glasgow, United Kingdom, 18-22 July 2022","authors":"","doi":"10.48550/arXiv.2307.15519","DOIUrl":"https://doi.org/10.48550/arXiv.2307.15519","url":null,"abstract":"","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"32 5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78038394","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}
Cartesian differential categories come equipped with a differential combinator that formalizes the directional derivative from multivariable calculus. Cartesian differential categories provide a categorical semantics of the differential lambda-calculus and have also found applications in causal computation, incremental computation, game theory, differentiable programming, and machine learning. There has recently been a desire to provide a (coordinate-free) characterization of Jacobians and gradients in Cartesian differential categories. One's first attempt might be to consider Cartesian differential categories which are Cartesian closed, such as models of the differential lambda-calculus, and then take the curry of the derivative. Unfortunately, this approach excludes numerous important examples of Cartesian differential categories such as the category of real smooth functions. In this paper, we introduce linearly closed Cartesian differential categories, which are Cartesian differential categories that have an internal hom of linear maps, a bilinear evaluation map, and the ability to curry maps which are linear in their second argument. As such, the Jacobian of a map is defined as the curry of its derivative. Many well-known examples of Cartesian differential categories are linearly closed, such as, in particular, the category of real smooth functions. We also explain how a Cartesian closed differential category is linearly closed if and only if a certain linear idempotent on the internal hom splits. To define the gradient of a map, one must be able to define the transpose of the Jacobian, which can be done in a Cartesian reverse differential category. Thus, we define the gradient of a map to be the curry of its reverse derivative and show this equals the transpose of its Jacobian. We also explain how a linearly closed Cartesian reverse differential category is precisely a linearly closed Cartesian differential category with an appropriate notion of transpose.
{"title":"Jacobians and Gradients for Cartesian Differential Categories","authors":"J. Lemay","doi":"10.4204/EPTCS.372.3","DOIUrl":"https://doi.org/10.4204/EPTCS.372.3","url":null,"abstract":"Cartesian differential categories come equipped with a differential combinator that formalizes the directional derivative from multivariable calculus. Cartesian differential categories provide a categorical semantics of the differential lambda-calculus and have also found applications in causal computation, incremental computation, game theory, differentiable programming, and machine learning. There has recently been a desire to provide a (coordinate-free) characterization of Jacobians and gradients in Cartesian differential categories. One's first attempt might be to consider Cartesian differential categories which are Cartesian closed, such as models of the differential lambda-calculus, and then take the curry of the derivative. Unfortunately, this approach excludes numerous important examples of Cartesian differential categories such as the category of real smooth functions. In this paper, we introduce linearly closed Cartesian differential categories, which are Cartesian differential categories that have an internal hom of linear maps, a bilinear evaluation map, and the ability to curry maps which are linear in their second argument. As such, the Jacobian of a map is defined as the curry of its derivative. Many well-known examples of Cartesian differential categories are linearly closed, such as, in particular, the category of real smooth functions. We also explain how a Cartesian closed differential category is linearly closed if and only if a certain linear idempotent on the internal hom splits. To define the gradient of a map, one must be able to define the transpose of the Jacobian, which can be done in a Cartesian reverse differential category. Thus, we define the gradient of a map to be the curry of its reverse derivative and show this equals the transpose of its Jacobian. We also explain how a linearly closed Cartesian reverse differential category is precisely a linearly closed Cartesian differential category with an appropriate notion of transpose.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"7 1","pages":"29-42"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90993593","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}
I present a formal connection between algebraic effects and game semantics, two important lines of work in programming languages semantics with applications in compositional software verification. Specifically, the algebraic signature enumerating the possible side-effects of a computation can be read as a game, and strategies for this game constitute the free algebra for the signature in a category of complete partial orders (cpos). Hence, strategies provide a convenient model of computations with uninterpreted side-effects. In particular, the operational flavor of game semantics carries over to the algebraic context, in the form of the coincidence between the initial algebras and the terminal coalgebras of cpo endofunctors. Conversely, the algebraic point of view sheds new light on the strategy constructions underlying game semantics. Strategy models can be reformulated as ideal completions of partial strategy trees (free dcpos on the term algebra). Extending the framework to multi-sorted signatures would make this construction available for a large class of games.
{"title":"Grounding Game Semantics in Categorical Algebra","authors":"Jérémie Koenig","doi":"10.4204/EPTCS.372.26","DOIUrl":"https://doi.org/10.4204/EPTCS.372.26","url":null,"abstract":"I present a formal connection between algebraic effects and game semantics, two important lines of work in programming languages semantics with applications in compositional software verification. Specifically, the algebraic signature enumerating the possible side-effects of a computation can be read as a game, and strategies for this game constitute the free algebra for the signature in a category of complete partial orders (cpos). Hence, strategies provide a convenient model of computations with uninterpreted side-effects. In particular, the operational flavor of game semantics carries over to the algebraic context, in the form of the coincidence between the initial algebras and the terminal coalgebras of cpo endofunctors. Conversely, the algebraic point of view sheds new light on the strategy constructions underlying game semantics. Strategy models can be reformulated as ideal completions of partial strategy trees (free dcpos on the term algebra). Extending the framework to multi-sorted signatures would make this construction available for a large class of games.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"11 3 1","pages":"368-383"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80393796","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 our earlier work on the compositional structure of cybernetic systems in order to account for the embodiment of such systems. All their interactions proceed through their bodies' boundaries: sensations impinge on their surfaces, and actions correspond to changes in their configurations. We formalize this morphological perspective using polynomial functors. The 'internal universes' of systems are shown to constitute an indexed category of statistical games over polynomials; their dynamics form an indexed category of behaviours. We characterize 'active inference doctrines' as indexed functors between such categories, resolving a number of open problems in our earlier work, and pointing to a formalization of the 'free energy principle' as adjoint to such doctrines. We illustrate our framework through fundamental examples from biology, including homeostasis, morphogenesis, and autopoiesis, and suggest a formal connection between spatial navigation and the process of proof.
{"title":"Polynomial Life: the Structure of Adaptive Systems","authors":"T. S. C. Smithe","doi":"10.4204/EPTCS.372.10","DOIUrl":"https://doi.org/10.4204/EPTCS.372.10","url":null,"abstract":"We extend our earlier work on the compositional structure of cybernetic systems in order to account for the embodiment of such systems. All their interactions proceed through their bodies' boundaries: sensations impinge on their surfaces, and actions correspond to changes in their configurations. We formalize this morphological perspective using polynomial functors. The 'internal universes' of systems are shown to constitute an indexed category of statistical games over polynomials; their dynamics form an indexed category of behaviours. We characterize 'active inference doctrines' as indexed functors between such categories, resolving a number of open problems in our earlier work, and pointing to a formalization of the 'free energy principle' as adjoint to such doctrines. We illustrate our framework through fundamental examples from biology, including homeostasis, morphogenesis, and autopoiesis, and suggest a formal connection between spatial navigation and the process of proof.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"5 1","pages":"133-148"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78427855","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 present categories of open dynamical systems with general time evolution as categories of coalgebras opindexed by polynomial interfaces, and show how this extends the coalgebraic framework to capture common scientific applications such as ordinary differential equations, open Markov processes, and random dynamical systems. We then extend Spivak's operad Org to this setting, and construct associated monoidal categories whose morphisms represent hierarchical open systems; when their interfaces are simple, these categories supply canonical comonoid structures. We exemplify these constructions using the 'Laplace doctrine', which provides dynamical semantics for active inference, and indicate some connections to Bayesian inversion and coalgebraic logic.
{"title":"Open dynamical systems as coalgebras for polynomial functors, with application to predictive processing","authors":"T. S. C. Smithe","doi":"10.4204/EPTCS.380.18","DOIUrl":"https://doi.org/10.4204/EPTCS.380.18","url":null,"abstract":"We present categories of open dynamical systems with general time evolution as categories of coalgebras opindexed by polynomial interfaces, and show how this extends the coalgebraic framework to capture common scientific applications such as ordinary differential equations, open Markov processes, and random dynamical systems. We then extend Spivak's operad Org to this setting, and construct associated monoidal categories whose morphisms represent hierarchical open systems; when their interfaces are simple, these categories supply canonical comonoid structures. We exemplify these constructions using the 'Laplace doctrine', which provides dynamical semantics for active inference, and indicate some connections to Bayesian inversion and coalgebraic logic.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"22 1","pages":"307-330"},"PeriodicalIF":0.0,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83916646","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 wide variety of bidirectional data accessors, ranging from mixed optics to functor lenses, can be formalized within a unique framework-dependent optics. Starting from two indexed categories, which encode what maps are allowed in the forward and backward directions, we define the category of dependent optics and establish under what assumptions it has coproducts. Different choices of indexed categories correspond to different families of optics: we discuss dependent lenses and prisms, as well as closed dependent optics. We introduce the notion of Tambara representation and use it to classify contravariant functors from the category of optics, thus generalizing the profunctor encoding of optics to the dependent case.
{"title":"Dependent Optics","authors":"Pietro Vertechi","doi":"10.4204/EPTCS.380.8","DOIUrl":"https://doi.org/10.4204/EPTCS.380.8","url":null,"abstract":"A wide variety of bidirectional data accessors, ranging from mixed optics to functor lenses, can be formalized within a unique framework-dependent optics. Starting from two indexed categories, which encode what maps are allowed in the forward and backward directions, we define the category of dependent optics and establish under what assumptions it has coproducts. Different choices of indexed categories correspond to different families of optics: we discuss dependent lenses and prisms, as well as closed dependent optics. We introduce the notion of Tambara representation and use it to classify contravariant functors from the category of optics, thus generalizing the profunctor encoding of optics to the dependent case.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"110 1","pages":"128-144"},"PeriodicalIF":0.0,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74481451","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}
Pub Date : 2022-01-01DOI: 10.48550/arXiv.2205.03906
B. Shapiro, David I. Spivak
Natural organized systems adapt to internal and external pressures and this seems to happens all the way down. Wanting to think clearly about this idea motivates our paper, and so the idea is elaborated extensively in the introduction, which should be broadly accessible to a philosophically-interested audience. In the remaining sections, we turn to more compressed category theory. We define the monoidal double category O rg of dynamic organizations, we provide definitions of O rg -enriched, or dynamic , categorical structures—e.g. dynamic categories, operads, and monoidal categories—and we show how they instantiate the motivating philo-sophical ideas. We give two examples of dynamic categorical structures: prediction markets as a dynamic operad and deep learning as a dynamic monoidal category.
{"title":"Dynamic categories, dynamic operads: From deep learning to prediction markets","authors":"B. Shapiro, David I. Spivak","doi":"10.48550/arXiv.2205.03906","DOIUrl":"https://doi.org/10.48550/arXiv.2205.03906","url":null,"abstract":"Natural organized systems adapt to internal and external pressures and this seems to happens all the way down. Wanting to think clearly about this idea motivates our paper, and so the idea is elaborated extensively in the introduction, which should be broadly accessible to a philosophically-interested audience. In the remaining sections, we turn to more compressed category theory. We define the monoidal double category O rg of dynamic organizations, we provide definitions of O rg -enriched, or dynamic , categorical structures—e.g. dynamic categories, operads, and monoidal categories—and we show how they instantiate the motivating philo-sophical ideas. We give two examples of dynamic categorical structures: prediction markets as a dynamic operad and deep learning as a dynamic monoidal category.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"1 1","pages":"183-202"},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87641977","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}
Pub Date : 2022-01-01DOI: 10.48550/arXiv.2211.01102
{"title":"Proceedings of the Fourth International Conference on Applied Category Theory, ACT 2021, Cambridge, United Kingdom, 12-16th July 2021","authors":"","doi":"10.48550/arXiv.2211.01102","DOIUrl":"https://doi.org/10.48550/arXiv.2211.01102","url":null,"abstract":"","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83425194","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 define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within G"ardenfors' framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these are the most general choice satisfying a criterion due to G"ardenfors and which are well-behaved compositionally. We then generalise these to define the category of log-concave probabilistic channels between convex spaces, which allows one to model fuzzy reasoning with noisy inputs, and provides a novel example of a Markov category.
{"title":"A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces","authors":"S. Tull","doi":"10.4204/EPTCS.372.22","DOIUrl":"https://doi.org/10.4204/EPTCS.372.22","url":null,"abstract":"We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within G\"ardenfors' framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these are the most general choice satisfying a criterion due to G\"ardenfors and which are well-behaved compositionally. We then generalise these to define the category of log-concave probabilistic channels between convex spaces, which allows one to model fuzzy reasoning with noisy inputs, and provides a novel example of a Markov category.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"74 1","pages":"306-322"},"PeriodicalIF":0.0,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83186596","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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