Pub Date : 2023-10-30DOI: 10.32408/compositionality-5-10
Masahito Hasegawa, Jean-Simon Pacaud Lemay
A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore category. The main theorem of this paper is that a symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad. We provide many examples of trace-coherent Hopf monads, such as those induced by cocommutative Hopf algebras or any symmetric Hopf monad on a compact closed category. We also explain how for traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator, while for traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We also provide separating examples of traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent.
{"title":"Traced Monads and Hopf Monads","authors":"Masahito Hasegawa, Jean-Simon Pacaud Lemay","doi":"10.32408/compositionality-5-10","DOIUrl":"https://doi.org/10.32408/compositionality-5-10","url":null,"abstract":"A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore category. The main theorem of this paper is that a symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad. We provide many examples of trace-coherent Hopf monads, such as those induced by cocommutative Hopf algebras or any symmetric Hopf monad on a compact closed category. We also explain how for traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator, while for traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We also provide separating examples of traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent.","PeriodicalId":500901,"journal":{"name":"Compositionality","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136022833","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 : 2023-10-04DOI: 10.32408/compositionality-5-9
Bolt, Joe, Hedges, Jules, Zahn, Philipp
This paper generalises the treatment of compositional game theory as introduced by Ghani et al. in 2018, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the notion of a game in the work by Ghani et al. is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issues all at once.
{"title":"Bayesian open games","authors":"Bolt, Joe, Hedges, Jules, Zahn, Philipp","doi":"10.32408/compositionality-5-9","DOIUrl":"https://doi.org/10.32408/compositionality-5-9","url":null,"abstract":"This paper generalises the treatment of compositional game theory as introduced by Ghani et al. in 2018, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the notion of a game in the work by Ghani et al. is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issues all at once.","PeriodicalId":500901,"journal":{"name":"Compositionality","volume":"223 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547523","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 categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality !L*, which has a limited edition of the contraction and permutation rules. The categorical part of the semantics is a monoidal biclosed category with a coalgebra modality, very similar to the structure of a Differential Category. We instantiate this category to finite dimensional vector spaces and linear maps via "quantisation" functors and work with three concrete interpretations of the coalgebra modality. We apply the model to construct categorical and concrete semantic interpretations for the motivating example of !L*: the derivation of a phrase with a parasitic gap. The effectiveness of the concrete interpretations are evaluated via a disambiguation task, on an extension of a sentence disambiguation dataset to parasitic gap phrases, using BERT, Word2Vec, and FastText vectors and Relational tensors.
{"title":"Categorical Vector Space Semantics for Lambek Calculus with a Relevant Modality","authors":"McPheat, Lachlan, Sadrzadeh, Mehrnoosh, Wazni, Hadi, Wijnholds, Gijs","doi":"10.32408/compositionality-5-2","DOIUrl":"https://doi.org/10.32408/compositionality-5-2","url":null,"abstract":"We develop a categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality !L*, which has a limited edition of the contraction and permutation rules. The categorical part of the semantics is a monoidal biclosed category with a coalgebra modality, very similar to the structure of a Differential Category. We instantiate this category to finite dimensional vector spaces and linear maps via \"quantisation\" functors and work with three concrete interpretations of the coalgebra modality. We apply the model to construct categorical and concrete semantic interpretations for the motivating example of !L*: the derivation of a phrase with a parasitic gap. The effectiveness of the concrete interpretations are evaluated via a disambiguation task, on an extension of a sentence disambiguation dataset to parasitic gap phrases, using BERT, Word2Vec, and FastText vectors and Relational tensors.","PeriodicalId":500901,"journal":{"name":"Compositionality","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135960552","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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