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### Preservation of Equations by Monoidal Monads

Louis Parlant, Jurriaan Rot, Alexandra Silva, Bas Westerbaan, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science
It has already been shown that any monoidal monad preserves linear equations; affine monads preserve drop equations (where some variable appears only on one side, such as x⋅ y = y) and relevant monads  ...  Finally, we identify a subclass of equations such that their preservation is equivalent to relevance.  ...  Later, Manes and Mulry show in [14] and [15] that a correspondence between categorical properties of one monad and algebraic features of another may lead to a distributive law.  ...

### Quotients in monadic programming: Projective algebras are equivalent to coalgebras [article]

Dusko Pavlovic, Peter-Michael Seidel
2017 arXiv   pre-print
of projective algebras is equivalent with the category of coalgebras for the comonad induced by any monad resolution.  ...  Both equivalences also entail several general corollaries concerning monadicity and comonadicity.  ...  1 Introduction The story Background: Monadic programming. Monads are one of functional programmers' favorite tools, and possibly one of the most popular categorical concepts [14, 13] .  ...

### Homotopy Theory of T-algebras over Top-Cat ? [article]

Ilias Amrani
2012 arXiv   pre-print
We discuss some properties and consequences of such path object. We also explain the construction of a 2-monad which algebras are (symmetric) monoidal topological categories.  ...  In this article, we interconnect two different aspects of higher category theory, in one hand the theory of infinity categories and on an other hand the theory of 2-categories.We construct an explicit  ...  Actually, this equality holds only in the case where r = s and z = v. Eventual model structure on T -algebras. This section si purely conjectural.  ...

### Polynomial monads and delooping of mapping spaces [article]

Michael Batanin, Florian De Leger
2020 arXiv   pre-print
small categories to polynomial monads and their algebras.  ...  As an application we give a categorical proof of the Dwyer-Hess and Turchin results concerning the explicit double delooping of spaces of long knots.  ...  The first author also gratefully acknowledges the financial support of Max Planck Institut für Mathematik. Bibliography.  ...

### Instances of Computational Effects: An Algebraic Perspective

Sam Staton
2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
We investigate the connections between computational effects, algebraic theories, and monads on functor categories.  ...  We use our framework to give a general account of several notions of computation that had previously been analyzed in terms of monads on presheaf categories: the analysis of local store by Plotkin and  ...  The development of the framework is guided by the categorical foundation of enriched algebraic theories [30] , building on recent developments by Lack and Rosický [32] , Melliès et al. ([37] , [38]  ...

### Preservation of Equations by Monoidal Monads [article]

Louis Parlant, Jurriaan Rot, Alexandra Silva, Bas Westerbaan
2020 arXiv   pre-print
It has already been shown that any monoidal monad preserves linear equations; affine monads preserve drop equations (where some variable appears only on one side, such as x· y = y) and relevant monads  ...  Finally, we identify the subclass of equations such that their preservation is equivalent to relevance.  ...  Later, Manes and Mulry show in [16] and [17] that a correspondence between categorical properties of one monad and algebraic features of another may lead to a distributive law.  ...

### Duality Theory and Categorical Universal Logic: With Emphasis on Quantum Structures

Yoshihiro Maruyama
2014 Electronic Proceedings in Theoretical Computer Science
Categorical Universal Logic is a theory of monad-relativised hyperdoctrines (or fibred universal algebras), which in particular encompasses categorical forms of both first-order and higher-order quantum  ...  We finally consider how to reconcile Birkhoff-von Neumann's quantum logic and Abramsky-Coecke's categorical quantum mechanics (which is modernised quantum logic as an antithesis to the traditional one)  ...  Both OML and EA are algebraic categories, i.e., can be described as categories of algebras of monads on Set.  ...

### On semiflexible, flexible and pie algebras

John Bourke, Richard Garner
2013 Journal of Pure and Applied Algebra
We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones.  ...  Clearly, to speak of an algebra for a 2-monad's being "free at the level of objects" is vague; and even if made precise, it would be insufficiently general, limiting us to 2-monads on Cat and similarly  ...  In the case that K is cocomplete, we may view [J, K] as the algebras for a 2-monad on C = [ob J, K], and now the pointwise equivalences therein are equally well the algebra maps which become equivalences  ...

### On semiflexible, flexible and pie algebras [article]

John Bourke, Richard Garner
2011 arXiv   pre-print
We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones.  ...  Pie algebras are contrasted with flexible and semiflexible algebras via a series of characterisations of each class; particular attention is paid to the case of pie, flexible and semiflexible weights,  ...  In the case that K is cocomplete, we may view [J, K] as the algebras for a 2-monad on C = [ob J, K], and now the pointwise equivalences therein are equally well the algebra maps which become equivalences  ...

### A Categorical Construction of the Real Unit Interval [article]

John van de Wetering
2021 arXiv   pre-print
The algebras of this monad carry an order, multiplication, addition and complement, and as such model much of the operations we need to do on probabilities.  ...  We show that the real unit interval is the unique non-initial, non-final irreducible algebra of a particular monad on the category of bounded posets.  ...  Acknowledgements The author would like to thank Bas and Bram Westerbaan for insightful discussions. The author is supported by a Rubicon fellowship financed by the Dutch Research Council (NWO).  ...

### Heterogeneous substitution systems revisited [article]

Benedikt Ahrens, Ralph Matthes
2016 arXiv   pre-print
we develop the proofs of the results of the cited paper and our new ones in UniMath, a recent library of univalent mathematics formalized in the Coq theorem prover.  ...  Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical description of substitution systems capable of capturing syntax involving binding which is independent of whether the syntax is  ...  Thanks to Paige North for discussion of the subject matter, and to Anders Mörtberg for providing feedback to a draft of this article.  ...

Chris Heunen, Martti Karvonen
2015 Electronical Notes in Theoretical Computer Science
are reversible precisely when the monad is Frobenius; by characterizing the largest reversible subcategory of Eilenberg-Moore algebras; and by identifying the latter algebras as measurements in our leading  ...  We extend categorical semantics of monadic programming to reversible computing, by considering monoidal closed dagger categories: the dagger gives reversibility, whereas closure gives higher-order expressivity  ...  Each side of the Frobenius law (1) equals ( ) † • ; one of these equations is equivalent to (1) .  ...

### Extensional Universal Types for Call-by-Value [chapter]

2008 Lecture Notes in Computer Science
., β-equality and ηequality hold for not only values but all terms. We give monadic style categorical semantics, so that the results can be applied also to languages like Haskell.  ...  We give syntax and its sound and complete categorical semantics, and describe how we can construct concrete models with a variety of computational effects.  ...  Acknowledgements I would like to thank Masahito Hasegawa for many helpful discussions and comments on earlier drafts.  ...