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].

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.

Completions of Non-Symmetric Metric Spaces Via Enriched Categories

2009 ◽  
Vol 16 (1) ◽  
pp. 157-182
Author(s):  
Vincent Schmitt
Keyword(s):  
Category Theory ◽  
Metric Spaces ◽  
Enriched Category ◽  
Closed Category ◽  
Enriched Categories ◽  
Terminal Object ◽  
Base Category ◽  
Monoidal Closed Category ◽  
The One ◽  
Universal Completion

Abstract It is known from [Lawvere, Repr. Theory Appl. Categ. 1: 1–37 2002] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0, ∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0, ∞] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the so-called dcpo completion.

scholarly journals Proof of a Conjecture of S. Mac Lane

1996 ◽  
Vol 3 (61) ◽  
Author(s):  
Sergei Soloviev
Keyword(s):  
Linear Logic ◽  
Sufficient Conditions ◽  
Vector Spaces ◽  
Categorical Equivalence ◽  
Closed Category ◽  
Symmetric Monoidal Closed Category ◽  
Monoidal Closed Category ◽  
Coherence Theorem ◽  
Mac Lane

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (only this case was considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the "abstract coherence theorem" found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture proves also the correctness of the modified description.

scholarly journals Note on monoidal localisation

1973 ◽  
Vol 8 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Brian Day
Keyword(s):  
Category Theory ◽  
Monoidal Category ◽  
Main Application ◽  
Closed Category ◽  
Monoidal Closed Category ◽  
Induced Structure

If a class Z of morphisms in a monoidal category A is closed under tensoring with the objects of A then the category obtained by inverting the morphisms in Z is monoidal. We note the immediate properties of this induced structure. The main application describes monoidal completions in terms of the ordinary category completions introduced by Applegate and Tierney. This application in turn suggests a “change-of-universe” procedure for category theory based on a given monoidal closed category. Several features of this procedure are discussed.

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