scholarly journals Internal coalgebras in cocomplete categories: Generalizing the Eilenberg–Watts theorem

2020 ◽  
pp. 2150165 ◽  
Author(s):  
Laurent Poinsot ◽  
Hans E. Porst
Keyword(s):  
Left Adjoint ◽  
Adjoint Functors ◽  
Locally Presentable Category ◽  
Completeness Result ◽  
Lawvere Theory

The category of internal coalgebras in a cocomplete category [Formula: see text] with respect to a variety [Formula: see text] is equivalent to the category of left adjoint functors from [Formula: see text] to [Formula: see text]. This can be seen best when considering such coalgebras as finite coproduct preserving functors from [Formula: see text], the dual of the Lawvere theory of [Formula: see text], into [Formula: see text]: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of [Formula: see text] into [Formula: see text]. Since [Formula: see text]-coalgebras in the variety [Formula: see text] for rings [Formula: see text] and [Formula: see text] are nothing but left [Formula: see text]-, right [Formula: see text]-bimodules, the equivalence above generalizes the Eilenberg–Watts theorem and all its previous generalizations. By generalizing and strengthening Bergman’s completeness result for categories of internal coalgebras in varieties, we also prove that the category of coalgebras in a locally presentable category [Formula: see text] is locally presentable and comonadic over [Formula: see text] and, hence, complete in particular. We show, moreover, that Freyd’s canonical constructions of internal coalgebras in a variety define left adjoint functors. Special instances of the respective right adjoints appear in various algebraic contexts and, in the case where [Formula: see text] is a commutative variety, are coreflectors from the category [Formula: see text] into [Formula: see text].

scholarly journals Varieties of topological groups and left adjoint functors

1973 ◽  
Vol 16 (2) ◽  
pp. 220-227 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Left Adjoint ◽  
Topological Groups ◽  
Adjoint Functors ◽  
Free Topological Group

In [6] and [2] Markov and Graev introduced their respective concepts of a free topological group. Graev's concept is more general in the sense that every Markov free topological group is a Graev free topological group. In fact, if FG(X) is the Graev free topological group on a topological space X, then it is the Markov free topological group FM(Y) on some space Y if and only if X is disconnected. This, however, does not say how FG(X) and FM(X) are related.

scholarly journals Adjunctions and braided objects

2014 ◽  
Vol 13 (06) ◽  
pp. 1450019 ◽  
Author(s):  
Alessandro Ardizzoni ◽  
Claudia Menini
Keyword(s):  
Monoidal Category ◽  
Forgetful Functor ◽  
Left Adjoint ◽  
Tensor Algebra ◽  
Adjoint Functors ◽  
Base Category ◽  
The One ◽  
Braided Bialgebras

In this paper, we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular, we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behavior of these functors in the case when the base category is braided is also considered.

scholarly journals Justification announcements in discrete time. Part II: Frame definability results

2018 ◽  
Vol 27 (5) ◽  
pp. 671-692
Author(s):  
Grigory K Olkhovikov
Keyword(s):  
Discrete Time ◽  
Time Structure ◽  
Stit Logic ◽  
Completeness Result ◽  
Time Part

Abstract In Part I of this paper we presented a Hilbert-style system $\Sigma _D$ axiomatizing stit logic of justification announcements interpreted over models with discrete time structure. In this part, we prove three frame definability results for $\Sigma _D$ using three different definitions of a frame plus another version of completeness result.

scholarly journals Separation Axioms in Intuitionistic Fuzzy Topological Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Amit Kumar Singh ◽  
Rekha Srivastava
Keyword(s):  
Topological Space ◽  
Left Adjoint ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

In this paper we have studied separation axiomsTi,i=0,1,2in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functorsℬ:IF-Top→BF-Topand𝒟:BF-Top→IF-Topand observe that𝒟is left adjoint toℬ.

scholarly journals Closed categories generated by commutative monads

1971 ◽  
Vol 12 (4) ◽  
pp. 405-424 ◽  
Author(s):  
Anders Kock
Keyword(s):  
Adjoint Functors ◽  
Closed Category ◽  
Base Category

The notion of commutative monad was defined by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the front- and end-adjunctions are closed transformations. (The terms ‘Closed Category’ etc. are from the paper [2] by Eilenberg and Kelly). In particular, the monad itself is a ‘closed monad’; this fact was also proved in [4].

Completeness of category-based equational deduction

1995 ◽  
Vol 5 (1) ◽  
pp. 9-40 ◽  
Author(s):  
Răzvan Diaconescu
Keyword(s):  
Logic Programming ◽  
Model Theory ◽  
Constraint Logic Programming ◽  
Horn Clause ◽  
Abstract Model ◽  
Abstract Model Theory ◽  
Completeness Result ◽  
Semantic Treatment ◽  
Theory Framework ◽  
Deduction Modulo

Equational deduction is generalised within a category-based abstract model theory framework, and proved complete under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and valuations as model morphisms rather than functions. Applications include many- and order-sorted (conditional) equational logics, Horn clause logic, equational deduction modulo a theory, constraint logics, and more, as well as any possible combination among them. In the cases of equational deduction modulo a theory and of constraint logic the completeness result is new. One important consequence is an abstract version of Herbrand's Theorem, which provides an abstract model theoretic foundation for equational and constraint logic programming.

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