## In this not...

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@MISC{Bergman_inthis,

author = {George M. Bergman},

title = {In this not...},

year = {}

}

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### Abstract

Abstract. If C and D are varieties of algebras in the sense of general algebra, then by a representable functor C → D we understand a functor which, when composed with the forgetful functor D → Set, gives a representable functor in the classical sense; Freyd showed that these functors are determined by D-coalgebra objects of C. Let

### Citations

171 |
The algebraic theory of semigroups
- Clifford, Preston
- 1961
(Show Context)
Citation Context ...Calling these x0 and x1, we find that the resulting representable functor takes a set A to A×A, with operation β((a0, a1), (b0, b1)) = (a0, b1); in semigroup theorists’ language, a “rectangular band” =-=[12]-=-. Of course, the coalgebras described above are merely the final objects in their respective categories, not the only such objects. For instance, for D the subvariety of Binar determined by (24), a ra... |

131 |
Terminal coalgebras in well-founded set theory
- Barr
- 1993
(Show Context)
Citation Context ...n with a morphism A → F (A) is called, in that usage, an F -coalgebra. The existence of final coalgebras in that sense has also been studiedsCOLIMITS OF REPRESENTABLE ALGEBRA-VALUED FUNCTORS 337 [1], =-=[4]-=-, and, as we shall see, has a slight overlap with the concept studied in this note. Still another use of “coalgebra”, probably the earliest, which also has relations to these two, and is basic to the ... |

106 | Infinite Abelian Groups. Vol - Fuchs - 1973 |

65 | Accessible categories: the foundations of categorical model theory, in - Makkai, Paré - 1989 |

59 | Categories for the Working Mathematician, Springer-Verlag Grad - Lane - 1971 |

37 | Lattice theory (3rd edition - BIRKHOFF - 1967 |

26 |
Natural Dualities for the Working Algebraist
- Clark, Davey
(Show Context)
Citation Context ... We find, in particular, that the initial representable functor Boole → Boole is the identity. There are other dualities between varieties of algebras and structures with compact topologies; cf. [2], =-=[11]-=-. It would be worth seeing whether these also lead to useful results about coalgebras, equivalently, representable functors. It would also be worth examining whether some of the properties of varietie... |

24 |
Algebra valued functors in general and tensor products in particular
- Freyd
- 1966
(Show Context)
Citation Context ...ure on the representing A-object. If A not only has small coproducts but general small colimits, the D-valued functors that are representable in this sense are precisely those that have left adjoints =-=[14]-=-, [6, Theorem 9.3.6], [9, Theorem 8.14]. We shall write Coalg(A, D) for the category of D-coalgebras in A, taking for the morphisms R → S those morphisms |R| → |S| that make commuting squares with the... |

23 | spaces, Cambridge Studies - Johnstone, Stone - 1982 |

14 |
A final coalgebra theorem, in: Category Theory and Computer
- Aczel, Mendler
- 1989
(Show Context)
Citation Context ... given with a morphism A → F (A) is called, in that usage, an F -coalgebra. The existence of final coalgebras in that sense has also been studiedsCOLIMITS OF REPRESENTABLE ALGEBRA-VALUED FUNCTORS 337 =-=[1]-=-, [4], and, as we shall see, has a slight overlap with the concept studied in this note. Still another use of “coalgebra”, probably the earliest, which also has relations to these two, and is basic to... |

13 |
An Invitation to General Algebra and Universal Constructions, pub
- Bergman
- 1998
(Show Context)
Citation Context ...IMITS OF REPRESENTABLE ALGEBRA-VALUED FUNCTORS 401 But its importance, as above, comes out in the description of morphisms among these functors and coalgebras.) For details, the reader can see [9] or =-=[6]-=-; but let me sketch some elementary examples. Let R be the comonoid representing the functor that takes every monoid A to the monoid A × A op ; let R ′ represent the functor that takes every A to the ... |

9 |
Coalgebras over a commutative ring
- Barr
- 1974
(Show Context)
Citation Context ...d [9, §§29-32, §43]. And in fact, as the referee has pointed out, results analogous to some of those in this note were proved for coalgebras in that sense over thirty years ago, by a similar approach =-=[3]-=-.) The first step toward our results will be an easy observation. 1.1. Lemma. If A has small colimits, then so does Coalg(A, D); equivalently, Rep(A, D) has small limits. Moreover, the underlying A-ob... |

7 | Some results on embeddings of algebras, after de Bruijn and - Bergman |

6 |
Representable functors and operations on rings
- Tall, Wraith
- 1970
(Show Context)
Citation Context ... We will apply these results to several examples in Part II below. We end Part I by noting – 8. A few directions for further investigation. In [9, §§63-64], generalizing work of D. Tall and G. Wraith =-=[21]-=- [22, §9], I defined a “TWcomonad” on a variety C to be a representable functor F : C → C with a comonad structure (morphisms F → F F and F → IdC making appropriate diagrams commute). Actually, the ma... |

2 |
Arens and Irving Kaplansky, Topological representations of algebras, Trans
- Richard
- 1948
(Show Context)
Citation Context ... → D. We find, in particular, that the initial representable functor Boole → Boole is the identity. There are other dualities between varieties of algebras and structures with compact topologies; cf. =-=[2]-=-, [11]. It would be worth seeing whether these also lead to useful results about coalgebras, equivalently, representable functors. It would also be worth examining whether some of the properties of va... |

2 | Some category-theoretic ideas in algebra (a too-brief tour of algebraic structure, monads, and the adjoint tower - Bergman - 1975 |

2 | Infinite Galois theory, Stone spaces, and profinite groups, course handout
- Bergman
- 1997
(Show Context)
Citation Context ...ieties D, such as those of associative (unital or nonunital) rings, semigroups, monoids, groups, and distributive lattices (and, automatically, all subvarieties of these), it is known [16, VI.2.6-9], =-=[8]-=- that The topological D-algebras with Stone topologies are precisely the inverse limits of systems of finite D-algebras, with the topologies induced by the discrete topologies on those finite algebras... |

2 |
Cogroups and Co-rings in Categories of Associative
- Bergman, Hausknecht
- 1996
(Show Context)
Citation Context ...ionally, rings, was already familiar to algebraic geometers, the representable functors corresponding to the “affine algebraic groups and rings”). For a more recent exposition, see [6, §§9.1-9.4]. In =-=[9]-=- the structure of Coalg(C, D) is determined for many particular varieties C and D. (The term “coalgebra” is sometimes used for a more elementary concept: Given any functor F : Set → Set, a set A given... |

2 |
On monoids and their dual, Boletín de la Sociedad Matemática Mexicana (2) 3
- Kan
- 1958
(Show Context)
Citation Context ...inal coalgebras have initial algebras for underlying algebra. But other limits of coalgebras involving these varieties need not be trivial. Let us briefly examine products in Coalg(Group, Group). Kan =-=[17]-=- determined the structure of all comonoid objects in Group. These all extend uniquely to cogroup structures, so we shall describe his result as determining the cogroups in Group. The functors these re... |

1 | revised version of notes - Wraith - 1969 |