Adjointness between theories and strict theories

Hans-Jürgen Vogel

Discussiones Mathematicae - General Algebra and Applications (2003)

  • Volume: 23, Issue: 2, page 163-212
  • ISSN: 1509-9415

Abstract

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The categorical concept of a theory for algebras of a given type was foundet by Lawvere in 1963 (see [8]). Hoehnke extended this concept to partial heterogenous algebras in 1976 (see [5]). A partial theory is a dhts-category such that the object class forms a free algebra of type (2,0,0) freely generated by a nonempty set J in the variety determined by the identities ox ≈ o and xo ≈ o, where o and i are the elements selected by the 0-ary operation symbols. If the object class of a dhts-category forms even a monoid with unit element I and zero element O, then one has a strict partial theory. In this paper is shown that every J-sorted partial theory corresponds in a natural manner to a J-sorted strict partial theory via a strongly d-monoidal functor. Moreover, there is a pair of adjoint functors between the category of all J-sorted theories and the category of all corresponding J-sorted strict theories. This investigation needs an axiomatic characterization of the fundamental properties of the category Par of all partial function between arbitrary sets and this characterization leads to the concept of dhts- and dhth∇s-categories, respectively (see [5], [11], [13]).

How to cite

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Hans-Jürgen Vogel. "Adjointness between theories and strict theories." Discussiones Mathematicae - General Algebra and Applications 23.2 (2003): 163-212. <http://eudml.org/doc/287687>.

@article{Hans2003,
abstract = {The categorical concept of a theory for algebras of a given type was foundet by Lawvere in 1963 (see [8]). Hoehnke extended this concept to partial heterogenous algebras in 1976 (see [5]). A partial theory is a dhts-category such that the object class forms a free algebra of type (2,0,0) freely generated by a nonempty set J in the variety determined by the identities ox ≈ o and xo ≈ o, where o and i are the elements selected by the 0-ary operation symbols. If the object class of a dhts-category forms even a monoid with unit element I and zero element O, then one has a strict partial theory. In this paper is shown that every J-sorted partial theory corresponds in a natural manner to a J-sorted strict partial theory via a strongly d-monoidal functor. Moreover, there is a pair of adjoint functors between the category of all J-sorted theories and the category of all corresponding J-sorted strict theories. This investigation needs an axiomatic characterization of the fundamental properties of the category Par of all partial function between arbitrary sets and this characterization leads to the concept of dhts- and dhth∇s-categories, respectively (see [5], [11], [13]).},
author = {Hans-Jürgen Vogel},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {symmetric monoidal category; dhts-category; partial theory; adjoint functor; -category},
language = {eng},
number = {2},
pages = {163-212},
title = {Adjointness between theories and strict theories},
url = {http://eudml.org/doc/287687},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Hans-Jürgen Vogel
TI - Adjointness between theories and strict theories
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2003
VL - 23
IS - 2
SP - 163
EP - 212
AB - The categorical concept of a theory for algebras of a given type was foundet by Lawvere in 1963 (see [8]). Hoehnke extended this concept to partial heterogenous algebras in 1976 (see [5]). A partial theory is a dhts-category such that the object class forms a free algebra of type (2,0,0) freely generated by a nonempty set J in the variety determined by the identities ox ≈ o and xo ≈ o, where o and i are the elements selected by the 0-ary operation symbols. If the object class of a dhts-category forms even a monoid with unit element I and zero element O, then one has a strict partial theory. In this paper is shown that every J-sorted partial theory corresponds in a natural manner to a J-sorted strict partial theory via a strongly d-monoidal functor. Moreover, there is a pair of adjoint functors between the category of all J-sorted theories and the category of all corresponding J-sorted strict theories. This investigation needs an axiomatic characterization of the fundamental properties of the category Par of all partial function between arbitrary sets and this characterization leads to the concept of dhts- and dhth∇s-categories, respectively (see [5], [11], [13]).
LA - eng
KW - symmetric monoidal category; dhts-category; partial theory; adjoint functor; -category
UR - http://eudml.org/doc/287687
ER -

References

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  2. [2] L. Budach and H.-J. Hoehnke, Automaten und Funktoren, Akademie Verlag, Berlin 1975 (the part: 'Allgemeine Algebra der Automaten', by H.-J. Hoehnke). 
  3. [3] S. Eilenberg and G.M. Kelly, Closed categories, Proc. Conf. Categorical Algebra (La Jolla, 1965), Springer, New York 1966, 421-562. Zbl0192.10604
  4. [4] P.J. Higgins, Algebras with a scheme of operators, Math. Nachr. 27 (1963), 115-132. Zbl0117.25903
  5. [5] H.-J. Hoehnke, On partial algebras, Colloq. Soc. J. Bolyai, Vol. 29, 'Universal Algebra; Esztergom (Hungary) 1977', North-Holland, Amsterdam, 1981, 373-412. 
  6. [6] G.M. Kelly, On MacLane's conditions for coherence of natural associativities, commutativities, etc, J. Algebra 4 (1964), 397-402. Zbl0246.18008
  7. [7] G.M. Kelly and S. MacLane, Coherence in closed categories, + Erratum, Pure Appl. Algebra 1 (1971), 97-140, 219. Zbl0212.35001
  8. [8] F.W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869-872. Zbl0119.25901
  9. [9] S. MacLane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), no. 4, 28-46. 
  10. [10] J. Schreckenberger, Über die zu monoidalen Kategorien adjungierten Kronecker-Kategorien, Dissertation (A), Päd. Hochschule, Potsdam 1978. 
  11. [11] J. Schreckenberger, Über die Einbettung von dht-symmetrischen Kategorien in die Kategorie der partiellen Abbildungen zwischen Mengen, Preprint P-12/80, Zentralinst. f. Math., Akad. d. Wiss. d. DDR, Berlin 1980. Zbl0442.18002
  12. [12] J. Schreckenberger, Zur Theorie der dht-symmetrischen Kategorien, Disseration (B), Päd. Hochschule Potsdam, Math.-Naturwiss. Fak., Potsdam 1984. 
  13. [13] H.-J. Vogel, Eine kategorientheoretische Sprache zur Beschreibung von Birkhoff-Algebren, Report R-Math-06/84, Inst. f. Math., Akad. d. Wiss. d. DDR, Berlin 1984. Zbl0554.18004
  14. [14] H.-J. Vogel, On functors between dht∇ -symmetric categories, Discuss. Math.-Algebra & Stochastic Methods, 18 (1998), 131-147. Zbl0921.18005
  15. [15] H.-J. Vogel, On properties of dht∇ -symmetric categories, Contributions to General Algebra 11 (1999), 211-223. 

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