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Novel Uses of Category Theory in Modeling OOP [article]

Moez A. AbdelGawad
2017 arXiv   pre-print
An outline and summary of four new potential applications of category theory to OOP research are presented.  ...  These include (1) the use of operads to model Java subtyping, (2) the use of Yoneda's lemma and representable functors in the modeling of generic types in generic nominally-typed OOP, (3) using a combination  ...  Yoneda's Lemma, Representable Functors, and OO Generic Types.  ... 
arXiv:1709.08056v4 fatcat:ckxy2fuverelfkm5y2hwmffl5q

The Fuzzified Natural Transformation between Categorial Functors and Its Selected Categorial Aspects

Krystian Jobczyk
2020 Symmetry  
Moreover, a multi-fuzzy Yoneda's lemma is formulated and proved. Finally, some references of these constructions to coding theory are elucidated in last parts of the paper.  ...  The natural transformation constitutes one of the most important entity of category theory and it introduces a piece of sophisticated dynamism to the categorial structures.  ...  Acknowledgments: The author would like to thank to Antoni and Krzysiek for so many motivating discussions.  ... 
doi:10.3390/sym12091578 fatcat:tp2jozpk2bgmdkdryjprlkdvy4

A remark on Yoneda's Lemma [article]

Shoji Yokura
2017 arXiv   pre-print
Yoneda'e Lemma is about the canonical isomorphism of all the natural transformations from a given representable covariant (contravariant, reps.) functor (from a locally small category to the category of  ...  sets) to a covariant (contravariant, reps.) functor.  ...  YONEDA'S LEMMA The well-known Yoneda's lemmas about representable functors are the following: From now, for the sake of simplicity, we denote h A (X) = hom C (X, A) simply by [X, A] and similarly [A,  ... 
arXiv:1712.02064v1 fatcat:go3u73un7bdbnp5kynbaahn4ou

Towards a Readable Formalisation of Category Theory

Greg O'Keefe
2004 Electronical Notes in Theoretical Computer Science  
We formally develop category theory up to Yoneda's lemma, using Isabelle/HOL/Isar, and survey previous formalisations.  ...  Thanks also to the referees for their insightful remarks.  ...  Thanks to Michael Norrish, Raj Góre, Jeremy Dawson, Amnon Neeman, Marieke Huisman, Tjark Weber, Lockwood Morris and Roy Dyckhoff for clues and helpful emails.  ... 
doi:10.1016/j.entcs.2003.12.014 fatcat:c5pslpe3sjaubby3li36t34avi

The Yoneda isomorphism commutes with homology

George Peschke, Tim Van der Linden
2016 Journal of Pure and Applied Algebra  
We show that, for a right exact functor from an abelian category to abelian groups, Yoneda's isomorphism commutes with homology and, hence, with functor derivation.  ...  As an application, we develop semiabelian (higher) torsion theories and the associated theory of (higher) universal (central) extensions.  ...  It therefore belongs to the very foundations of the theory of categories.  ... 
doi:10.1016/j.jpaa.2015.05.005 fatcat:jc4gvp4vo5fjbic4brfd2twylm

Professor Nobuo Yoneda (28 March 1930–22 April 1996)

Eiiti Wada, Akinori Yonezawa
1996 Science of Computer Programming  
As a Mathematician, he proved a fundamental theorem known as Yoneda's Lemma in Category Theory, which is nowadays referred to in most standard textbooks and papers in this field.  ...  Dr Yoneda's activities were not confined to research and education.  ... 
doi:10.1016/0167-6423(96)88115-9 fatcat:qu2qzqws4vgulnpsysha6xzafm

Notes on A^1-contractibility and A^1-excision [article]

Yuri Shimizu
2018 arXiv   pre-print
We prove that a smooth scheme of dimension n over a perfect field is A^1-weakly equivalent to a point if it is A^1-n-connected.  ...  I would like to thank my adviser Shohei Ma for many useful advices. I would also like to thank Aravind Asok for helpful comments.  ...  A 1 -homotopy theory is constructed by using simplicial sets. We refer to [GJ] for the homotopy theory of simplicial sets. Let Spc k be the category of simplicial Nisnevich sheaves of sets.  ... 
arXiv:1809.01895v1 fatcat:odttqa25ivgmbadzktdxtq6jyy

Towards a Categorical Theory of Creativity for Music, Discourse, and Cognition [chapter]

Moreno Andreatta, Andrée Ehresmann, René Guitart, Guerino Mazzola
2013 Lecture Notes in Computer Science  
The model, which is applied to musical creativity, discourse theory, and cognition, suggests the relevance of the notion of "colimit" as a unifying construction in the three domains as well as the central  ...  role played by the Yoneda Lemma in the categorical formalization of creative processes.  ...  Given a category C, Yoneda's Lemma says that the knowledge of C ∈ C 0 is equivalent to the knowledge of @C.  ... 
doi:10.1007/978-3-642-39357-0_2 fatcat:hzquhy3warc5zpj5f5gl3wiwbq

Universal birational invariants and A^1-homology [article]

Yuri Shimizu
2020 arXiv   pre-print
In this paper, we prove that the functor of zeroth A^1-homology H^A^1_0 is universal as a functorial birational invariant of smooth proper k-varieties taking values in a category enriched by abelian groups  ...  I would like to thank my adviser Shohei Ma for many useful advices. I also would like to thank Tom Bachmann for a helpful comment. This work was supported by JSPS KAKENHI Grant Number JP19J21433.  ...  Applications to A 1 -homotopy theory In this section, we give some applications to A 1 -homotopy theory.  ... 
arXiv:2002.05918v2 fatcat:36m7ebnqebg55kgnjkbmts6ln4

Epimorphisms of additive categories up to direct factors

Henning Krause
2005 Journal of Pure and Applied Algebra  
We characterize additive functors which are epimorphisms up to direct factors.  ...  The assumption on F * implies that for each object M in (D, Ab), the natural map Using Yoneda's lemma, we find an element i i in i∈ Hom D (FX i , Y ) which is sent to id Y .  ...  It follows from Yoneda's lemma that the induced functor proj F is fully faithful if and only if F is fully faithful.  ... 
doi:10.1016/j.jpaa.2005.03.012 fatcat:5h6syisu3vcptjbyngeulp7vma

Quotients of one-sided trianglated categories by rigid subcategories as module categories [article]

Zengqiang Lin, Yang Zhang
2013 arXiv   pre-print
We prove that some subquotient categories of one-sided triangulated categories are abelian.  ...  This unifies a result by Iyama-Yoshino in the case of triangulated categories and a result by Demonet-Liu in the case of exact categories.  ...  Introduction Cluster tilting theory gives a way to construct abelian categories from some triangulated categories.  ... 
arXiv:1302.2062v1 fatcat:zsvi2qtelzbklgdtdzzwmgckdq

Relative 𝔸^1-homology and its applications [article]

Yuri Shimizu
2021 arXiv   pre-print
In order to prove these results, we develop a general theory of relative 𝔸^1-homology and 𝔸^1-homotopy sheaves.  ...  I would like to thank my adviser Shohei Ma for many  ...  the category of smooth k-schemes Sm k to DM ef f − (k).  ... 
arXiv:1904.08644v3 fatcat:3inlnxazaneg7csywrkeewba4y

Auslander–Reiten Theory via Brown Representability Dedicated to Professor Daniel Quillen on his Sixtieth Birthday

Henning Krause
2000 K-theory  
We develop an Auslander-Reiten theory for triangulated categories which is based on Brown's representability theorem. (2000) : 18E30, 55U35, 16G70, 18G25. Mathematics subject classifications  ...  Acknowledgement I would like to thank Apostolos Beligiannis and Idun Reiten for a number of helpful conversations about the subject of this paper.  ...  Note that I ∼ = Hom(C, Z) by Yoneda's lemma. Now let β : Y → Z be the map corresponding to the X and claim that α has the desired properties.  ... 
doi:10.1023/a:1026571214620 fatcat:mficj3wjenhhngcfznxpxufc2y

Excellent extensions of tilted algebras

Juxiang Sun
2019 International Journal of Algebra  
Lemma 2.4.(see [[16], For any additional category A we denote by (A op , Ab) the category of contravariant functors from A to Ab, where Ab is the category of all abelian groups.  ...  Hence, Hom A (−, M ) is projective in add T , and so M ∈ add T by Yoneda's lemma.  ... 
doi:10.12988/ija.2019.91040 fatcat:7rggknn52rhczesktk2trwhd54

On the Monomorphism Category of n-Cluster Tilting Subcategories [article]

Javad Asadollahi, Rasool Hafezi, Somayeh Sadeghi
2020 arXiv   pre-print
We construct two functors from S(M) to modM, the category of finitely presented (coherent) additive contravariant functors on the stable category of M.  ...  So they induce equivalences from the quotient categories of the submodule category of M modulo their respective kernels. Moreover, they are related by a syzygy functor on the stable category of modM.  ...  By Yoneda's lemma, we get a monomorphism M f → M ′ .  ... 
arXiv:2008.04178v1 fatcat:w3igeaczbjasbhmvy24fkakmkm
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