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Novel Uses of Category Theory in Modeling OOP
[article]
2017
arXiv
pre-print
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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
2020
Symmetry
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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]
2017
arXiv
pre-print
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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
2004
Electronical Notes in Theoretical Computer Science
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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
2016
Journal of Pure and Applied Algebra
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2015 pre-print arXiv:1505.05406v1 fatcat:pzc6akaatnc7vk4sswsh55qtyyOther Versions
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)
1996
Science of Computer Programming
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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]
2018
arXiv
pre-print
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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]
2013
Lecture Notes in Computer Science
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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]
2020
arXiv
pre-print
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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
2005
Journal of Pure and Applied Algebra
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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]
2013
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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
2000
K-theory
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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
2019
International Journal of Algebra
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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]
2020
arXiv
pre-print
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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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