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Covariant and contravariant approaches to topology

Jerzy Dydak
1997 International Journal of Mathematics and Mathematical Sciences  
The purpose of [2] is to present a way of viewing of basic topology which unifies quite a few results and concepts previously seemed not related (quotient maps, product topology, subspace topology, separation  ...  It is shown in [2] that one can introduce the PC-product X x PC Y and the CO-product X x co Y so that if Top {PC, CO}, then one has a natural equivalence 1 . 1 COVARIANT AND CONTRAVARIANT TOPOLOGIES  ...  Obviously, this particular functor Hom mst be yX with the compact-open topology (by the uniqueness of adjoint functors) but the approach of [2] is much better integrated with thc category theory and,  ... 
doi:10.1155/s0161171297000860 fatcat:v3bxfzpjp5h5xi436iirakrg6y

Relational Parametricity and Quotient Preservation for Modular (Co)datatypes [chapter]

Andreas Lochbihler, Joshua Schneider
2018 Lecture Notes in Computer Science  
Bounded natural functors (BNFs) provide a modular framework for the construction of (co)datatypes in higher-order logic.  ...  In this paper, we generalise BNFs such that the mapper and relator act on both covariant and contravariant parameters.  ...  The authors thank Dmitriy Traytel, Andrei Popescu, and the anonymous reviewers for inspiring discussions and suggestions how to improve the presentation. The authors are listed alphabetically.  ... 
doi:10.1007/978-3-319-94821-8_24 fatcat:kdp7abrlfngg3pbvysfis6wbey

Commuting Homotopy Limits and Smash Products

Wolfgang Lück, Holger Reich, Marco Varisco
2003 K-theory  
In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute.  ...  In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group.  ...  One easily checks the following lemma. which is natural in X, Y and Z. A covariant (contravariant) I-spectrum is a covariant (contravariant) functor E : I → SPECTRA.  ... 
doi:10.1023/b:kthe.0000018387.87156.c4 fatcat:przxmky5ancsbmfpvmuy2g2lui

General theory of natural equivalences

Samuel Eilenberg, Saunders MacLane
1945 Transactions of the American Mathematical Society  
. 2 Let T be a functor covariant in 2Í and contravariant in 33, with values in E.  ...  21 and ß:Bi-^>B2 in 33 functor covariant in 21, contravariant in 33, with values in E  ...  With the natural addition and topology, the set of all factor sets/ of II in G constitute a topological abelian group Fact (G, H).  ... 
doi:10.1090/s0002-9947-1945-0013131-6 fatcat:nabhm4jcnjgqjpgeptl7kmwdfi

Limits and colimits in generalized algebraic categories

Jiří Adámek
1976 Czechoslovak Mathematical Journal  
F covariant, G contravariant F contravariant, G covariant We investigate the existence of hmits and cohmits in the categories A(F, G) and A{G, F) where F is an arbitrary contravariant set functor and G  ...  Let Я be an arbitrary non-constant set-functor (covariant or contravariant).  ... 
doi:10.21136/cmj.1976.101372 fatcat:jfx63jxji5drpj2h4snvbwsv4q

On the algebraic L-theory of Δ-sets [article]

Andrew Ranicki
2010 arXiv   pre-print
The algebraic L-groups L_*(,X) are defined for an additive category with chain duality and a Δ-set X, and identified with the generalized homology groups H_*(X;_∙()) of X with coefficients in the algebraic  ...  An object in A * (X) is an induced contravariant functor F : X → A and the composite of the contravariant functors X F G G A T G G A is a covariant functor, let alone an induced contravariant functor.  ...  (i) The contravariant and covariant functor categories are related by the identities Definition 1.8.  ... 
arXiv:math/0701833v2 fatcat:4zur6bjhz5hz3iikskgfi37leu

Endofunctors of singularity categories characterizing Gorenstein rings

Takuma Aihara, Ryo Takahashi
2015 Proceedings of the American Mathematical Society  
Denote by D sg (Λ) the singularity category of Λ, that is, the Verdier quotient of the bounded derived category D b (Λ) of finitely generated (right) Λ-modules by the full subcategory consisting of bounded  ...  What contravariant endofunctor of D sg (Λ) characterizes the Iwanaga-Gorenstein property of Λ?  ...  The natural full embedding functor CM(R) → D b (R) induces an additive covariant functor η : CM(R) → D sg (R).  ... 
doi:10.1090/proc/12580 fatcat:2wokkyliwjauxaawhpk2fyh2se

Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K‐ and L‐Theory

James F. Davis, Wolfgang Lück
1998 K-theory  
Let E: Or(G) −→ SPECTRA be a covariant functor. We define a functor E % : G-SPACES −→ SPECTRA by setting E % (X) = (G/H −→ X H ) + ⊗ Or(G) E.  ...  By approximating a C-space X by a free C-CW -complex, we define in Section 4 homology H C * (X; E) and cohomology H * C (X; E) of a space X with coefficients in a C-spectrum E.  ...  The tensor product of a contravariant C-space with a covariant C space yields a topological space. DEFINITION 1.4 . Let X be a contravariant and Y be a covariant C-space.  ... 
doi:10.1023/a:1007784106877 fatcat:pmtheokq6ne6vnagb7rpxb7igy

Quantum Fields as Category Algebras [article]

Hayato Saigo
2021 arXiv   pre-print
We define quantum fields and their states as category algebras and states on causal categories with partial involution structures.  ...  Conceptual relationships with conventional approaches to quantum fields, including Algebraic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT), are also discussed.  ...  Misa Saigo and Mr. Juzo Nohmi for fruitful discussions and comments.  ... 
arXiv:2108.12936v5 fatcat:duuqrqrqsja55nqhty4nzn6nna

Quantum Fields as Category Algebras

Hayato Saigo
2021 Symmetry  
We define quantum fields and their states as category algebras and states on causal categories with partial involution structures.  ...  Conceptual relationships with conventional approaches to quantum fields, including Algebraic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT), are also be discussed.  ...  Acknowledgments: The author is grateful to Hiroshi Ando, Takahiro Hasebe, Soichiro Fujii, Izumi Ojima, Kazuya Okamura, Misa Saigo, and Juzo Nohmi for the fruitful discussions and comments.  ... 
doi:10.3390/sym13091727 fatcat:sskh7uyvnbb6hkssviw6htovni

Quantum energy inequalities and local covariance II: categorical formulation

Christopher J. Fewster
2007 General Relativity and Gravitation  
Covariant formulations of the numerical range and spectrum of locally covariant fields are given and investigated, and a new algebra of fields is identified, in which fields are treated independently of  ...  their realisation on particular spacetimes and manifestly covariant versions of the functional calculus may be formulated.  ...  Acknowledgements: The author thanks the Max Planck Institute for Mathematics in the Natural Sciences, Leipzig, and the Institute for Theoretical Physics, Göttingen, for hospitality during the course of  ... 
doi:10.1007/s10714-007-0494-3 fatcat:pmz2bnn56vcqhmh5c2yyehaoxm

Page 528 of Mathematical Reviews Vol. 55, Issue 2 [page]

1978 Mathematical Reviews  
Precisely, given a contravariant functor G, the dual functor G* is defined by G(X)=Nat(G, (X@ -)).  ...  In particular, *-reflexive functors are characterized as maximal subfunctors of contravariant hom functors H(-,A) with A reflexive, although the reader must be cautioned about the leitient sense in which  ... 

Inductive, coinductive, and pointed types

Brian T. Howard
1996 Proceedings of the first ACM SIGPLAN international conference on Functional programming - ICFP '96  
The motivations for this work are certain natural constructions in category theory, in particular the notion of an algebraically bounded functor, due to Freyd.  ...  the facilities for bounded iteration and coiteration.  ...  Next, we need to show that if F is a contravariant functor, then the fixpoints of F are the same as the fixpoints of the covariant functor F 2 , provided F 2 is algebraically bounded.  ... 
doi:10.1145/232627.232640 dblp:conf/icfp/Howard96 fatcat:obxv3kz5xbg3ndnoif7gfdcbua

Inductive, coinductive, and pointed types

Brian T. Howard
1996 SIGPLAN notices  
The motivations for this work are certain natural constructions in category theory, in particular the notion of an algebraically bounded functor, due to Freyd.  ...  the facilities for bounded iteration and coiteration.  ...  Next, we need to show that if F is a contravariant functor, then the fixpoints of F are the same as the fixpoints of the covariant functor F 2 , provided F 2 is algebraically bounded.  ... 
doi:10.1145/232629.232640 fatcat:bq6gf5faajhm7p24gt63bgwnqm

Variations on the bagdomain theme

P.T. Johnstone
1994 Theoretical Computer Science  
The purpose of the present paper is to introduce some of these possibilities and to describe their basic properties, using the theory of fibrations and partial products developed in (Johnstone, 1993) .  ...  The notion of bagdomain was first introduced by Vickers (1992) and further studied by the present author in (Johnstone, 1992) .  ...  We conclude with a simple example of a "mixed" bagdomain one that is both covariant and contravariant.  ... 
doi:10.1016/0304-3975(94)00120-8 fatcat:jk5y653l2zfqpg36ngtkc5e5x4
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