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A linear category of polynomial diagrams

PIERRE HYVERNAT
2013 Mathematical Structures in Computer Science  
We present a categorical model for intuitionistic linear logic where objects are polynomial diagrams and morphisms are simulation diagrams.  ...  ) structures require additional properties and are only developed in the category Set, where the objects and morphisms have natural interpretations in terms of games, simulation and strategies.  ...  Introduction Categories of games abound in the literature of denotational semantics of linear logic. We present a category based on a notion of game that differs from traditional games semantics.  ... 
doi:10.1017/s0960129512001016 fatcat:fmbn6clxlrhf3fofi7b5betsti

A Linear Category of Polynomial Functors (extensional part)

Pierre Hyvernat, Valeria de Paiva
2014 Logical Methods in Computer Science  
We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms).  ...  We then make this category into a model for intuitionistic linear logic by defining an additive and exponential structure.  ...  A LINEAR CATEGORY OF POLYNOMIAL FUNCTORS 5 P 2 ]] by a universal property relying only on [[P 1 ]] and [[P 2 ]], thus giving an extensional definition of the tensor of polynomial functors.  ... 
doi:10.2168/lmcs-10(2:2)2014 fatcat:ai65fezjtfcllhhx6clnrc4wre

Linearization and categorification [article]

Mikhail Khovanov
2016 arXiv   pre-print
We discuss the notion of linearization through examples, which include the Price map, PageRank, representation theory, the Euler characteristic and quantum invariants.  ...  We also review categorification, which adds an additional layer of structure, in the context of the last two examples.  ...  ) to elements of a linear structure, the ring of Laurent polynomials.  ... 
arXiv:1603.08223v1 fatcat:3ub7asuaevhwhhimucmtxc3wm4

Categorification of Hopf algebras of rooted trees

Joachim Kock
2013 Open Mathematics  
AbstractWe exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P.  ...  This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing  ...  Partial support from research grants MTM 2009-10359 and MTM 2010-20692 of Spain and 2009 SGR-1092 of Catalonia is gratefully acknowledged.  ... 
doi:10.2478/s11533-012-0152-1 fatcat:zibp44je7ba2zhg5ttahjweprm

The Singular Temperley-Lieb Category

Carmen Caprau, Joel Smith
2014 ISRN Geometry  
We also construct a couple of polynomial invariants for oriented tangles from category theory point of view.  ...  Our construction is motivated by a state model for the sl(2) polynomial of an oriented link and provides a categorical perspective to this link invariant.  ...  Acknowledgment During the research of this paper, the second author was partially supported by the California State University, Fresno, CA, USA, through an undergraduate research grant.  ... 
doi:10.1155/2014/321509 fatcat:wq4vcqzopzdmle67ooyeosrixi

Monads in Double Categories [article]

Thomas M. Fiore, Nicola Gambino, Joachim Kock
2010 arXiv   pre-print
In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means for a double category to admit the construction of free monads.  ...  Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does.  ...  We gratefully acknowledge the support and hospitality of the Centre de Recerca Matemàtica during the academic year 2007/08, in occasion of the thematic programme on Homotopy Theory and Higher Categories  ... 
arXiv:1006.0797v2 fatcat:apnsxknf6fdmxpcojwcvqujapi

Functorial aggregation [article]

David I. Spivak
2022 arXiv   pre-print
Starting with the category 𝐏𝐨𝐥𝐲 of polynomial functors in one variable, we review the relatively recent results of Ahman-Uustalu and Garner, which showed that the framed bicategory ℂ𝐚𝐭^♯ of comonads  ...  But whereas querying has an elegant category-theoretic treatment in terms of parametric right adjoints between copresheaf categories, a categorical formulation of aggregation – especially one that lives  ...  We refer to a comonoid or bicomodule as linear if its carrier polynomial is linear.  ... 
arXiv:2111.10968v4 fatcat:kzk2x2dvynffrkhvs5rsqewoze

Centrifugal Compressor Polytropic Performance—Improved Rapid Calculation Results—Cubic Polynomial Methods

Matt Taher, Fred Evans
2021 International Journal of Turbomachinery, Propulsion and Power  
A broad range of example case results verify the accuracy and ease of use of the method.  ...  This paper presents a new improved approach to calculation of polytropic performance of centrifugal compressors.  ...  compression path on T-s and h-s diagrams to a high degree of accuracy.  ... 
doi:10.3390/ijtpp6020015 fatcat:k3emnesh5zhj5luzytjxv77ujm

Monads in double categories

Thomas M. Fiore, Nicola Gambino, Joachim Kock
2011 Journal of Pure and Applied Algebra  
In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means for a double category to admit the construction of free monads.  ...  Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does.  ...  We gratefully acknowledge the support and hospitality of the Centre de Recerca Matemàtica during the academic year 2007/08, on the occasion of the thematic programme on Homotopy Theory and Higher Categories  ... 
doi:10.1016/j.jpaa.2010.08.003 fatcat:hmhd2i6q3bhdvnixgf5fxwjqpq

Categorical Complexity [article]

Saugata Basu, M. Umut Isik
2019 arXiv   pre-print
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity  ...  We discuss several examples of this new definition in categories of wide common interest, such as finite sets, Boolean functions, topological spaces, vector spaces, semi-linear and semi-algebraic sets,  ...  (D) The category SL (respectively, SA) of embedded semi-linear (respectively, semialgebraic) sets and affine (respectively, polynomial) maps.  ... 
arXiv:1610.07737v4 fatcat:7fblz6baknhtddjridnz6uoj5e

Comparing the orthogonal and unitary functor calculi [article]

Niall Taggart
2021 arXiv   pre-print
We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus  ...  The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces.  ...  Consider the diagram of enriched categories (n)E U n → U(n)E O 2n is a right Quillen functor. This produced a diagram of adjoint functors Sp U [U(n)] r !  ... 
arXiv:2001.04485v2 fatcat:jhwnhdns4zgltkmqlbrm7v3zwq

Prediction of pilot induced oscillations

PANĂ Valentin
2011 INCAS Bulletin  
The paper presents a method to detect possible pilot-induced oscillations of Category II (with rate and position limiting), a phenomenon usually due to a misadaptation between the pilot and the aircraft  ...  In this analysis the nonlinear elements are substituted by linear uncertain parameters. This approach assumes that PIO are characterized by a limit cycle behavior.  ...  Fig. 5 -LFig. 6 - 56 Block diagram of the X-15 pilot vehicle systemThe numerical values of the elements in the block diagram are: 6 is presented the result of Robust Stability Analysis versus a time-invariant  ... 
doi:10.13111/2066-8201.2011.3.1.9 fatcat:nfi4v67frnhanbvooznl3zgr3m

Resolutions of ideals associated to subspace arrangements [article]

Francesca Gandini
2019 arXiv   pre-print
To prove these results we rely on the functoriality of free resolutions and construct a functor Ω from the category of polynomial functors to itself.  ...  As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal.  ...  Lemma 3 . 3 A polynomial functor P on the category of finite dimensional vector spaces Vec induces a functor P V on the category of GL(V )-representations Rep V .  ... 
arXiv:1810.11694v3 fatcat:vg7q4ogugbev5mebbqtu27qgsy

Recursion Formulas for HOMFLY and Kauffman Invariants [article]

Qingtao Chen, Nicolai Reshetikhin
2014 arXiv   pre-print
In this note we describe the recursion relations between two parameter HOMLFY and Kauffman polynomials of framed links These relation correspond to embeddings of quantized universal enveloping algebras  ...  For a ring A, define T an ′ A as the additive Alinear category where objects are direct sum of objects in T an ′ and morphisms are linear combination of morphisms from T an ′ .  ...  For a ring A, define the additive category T an A as the category with objects being direct sums of objects of T an but with morphisms being Alinear combination of tangles.  ... 
arXiv:1401.1927v1 fatcat:kwbbktzhbvheph6izgrb5x2xmy

Comparing orthogonal calculus and calculus with Reality [article]

Niall Taggart
2022 arXiv   pre-print
We show that there exists a suitable C_2-fixed points functor from calculus with Reality to the orthogonal calculus of Weiss which recovers orthogonal calculus "up to a shift" in an analogous way with  ...  the recovery of real topological K-theory from Atiyah's K-theory with Reality via appropriate C_2-fixed points.  ...  linear isometries to the category of (based) spaces.  ... 
arXiv:2205.15660v1 fatcat:jwpa46todvb65edzmmaa46hwza
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