Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Algebraic Theories
  • Cited by 17
Publisher:
Cambridge University Press
Online publication date:
June 2011
Print publication year:
2010
Online ISBN:
9780511760754
DOI:
https://doi.org/10.1017/CBO9780511760754
Subjects:
Mathematics (general), Mathematics, Logic, Categories and Sets, Algebra
Series:
Cambridge Tracts in Mathematics (184)
Information

Book description

Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area.

Reviews

'The book is very well written and made as self-contained as it is reasonable for the intended audience of graduate students and researchers.'

Source: Zentralblatt MATH

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Page 1 of 2

Contents

Page 1 of 2

References
References
References
Adámek, J. (1974). Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolina. 14: 589–602.
Adámek, J. (1977). Colimits of algebras revisited. Bull. Austral. Math. Soc. 17: 433–450.
Adámek, J., F., Borceux, S., Lack, and J., Rosický (2002). A classification of accessible categories. J. Pure Appl. Algebra 175: 7–30.
Adámek, J., H., Herrlich, and G., Strecker (2009). Abstract and Concrete Categories. Dover.
Adámek, J., F. W., Lawvere, and J., Rosický (2001a). How algebraic is algebra? Theory Appl. Categ. 8: 253–283.
Adámek, J., F. W., Lawvere, and J., Rosický (2003). On the duality between varieties and algebraic theories. Alg. Univ. 49: 35–49.
Adámek, J., and H.-E., Porst (1998). Algebraic theories of quasivarieties. J. Algebra 208: 379–398.
Adámek, J., and J., Rosický (1994). Locally Presentable and Accessible Categories. Cambridge University Press.
Adámek, J., and J., Rosický (2001). On sifted colimits and generalized varieties. Theory Appl. Categ. 8: 33–53.
Adámek, J., J., Rosický, and E. M., Vitale (2001b). On algebraically exact categories and essential localizations of varieties. J. Algebra 244: 450–477.
Adámek, J., J., Rosický, and E. M., Vitale (2010). What are sifted colimits? Theory Appl. Categ. 23: 251–260
Adámek, J., M., Sobral, and L., Sousa (2006). Morita equivalence for many-sorted algebraic theories. J. Algebra 297: 361–371.
Almeida, J. (1994). Finite Semigroups and Universal Algebra. World Scientific.
Artin, M., A., Grothendieck, and J. L., Verdier (1972). Théorie des topos et cohomologie étale des schémas. Lect. Notes in Math. 269. Springer.
Banaschewski, B. (1972). Functors into categories of M-sets. Abh. Mat. Sem. Univ. Hamburg 38: 49–64.
Barr, M. (1970). Coequalizers and free triples. Math. Z. 116: 307–322.
Barr, M., A. P., Grillet, and D. H., van Osdol (1971). Exact Categories and Categories of Sheaves. Lect. Notes in Math. 236. Springer.
Barr, M., and C., Wells (1985). Toposes, Triples and Theories. Springer.
Barr, M., and C., Wells (1990). Category Theory for Computing Science. Prentice Hall.
Bass, H. (1968). Algebraic K-Theory. Benjamin.
Bastiani, A., and C., Ehresmann (1972). Categories of sketched structures. Cah. Top. Géom. Différ. Catég. 13: 104–214.
Bénabou, J. (1967). Introduction to bicategories. In Reports of the Midwest Category Seminar. Lect. Notes in Math. 40. Springer, 1–77.
Bénabou, J. (1968). Structures algébriques dans les catégories. Cah. Top. Géom. Différ. Catég. 10: 1–126.
Birkhoff, G. (1935). On the structure of abstract algebras. Proc. Cambr. Phil. Soc. 31: 433–454.
Birkhoff, G., and J. D., Lipson (1970). Heterogenous algebras. J. Combin. Theory 8: 115–133.
Boardman, J. M., and R. M., Vogt (1973). Homotopy Invariant Algebraic Structures on Topological Spaces. Lect. Notes in Math. 347. Springer.
Borceux, F. (1994). Handbook of Categorical Algebra. Cambridge University Press.
Borceux, F., and E. M., Vitale (1994). On the notion of bimodel for functorial semantics. Appl. Categ. Struct. 2: 283–295.
Bourbaki, N. (1956). Théorie des Ensembles. Herrman.
Bunge, M. (1966). Categories of set-valued functors. Dissertation, University of Pennsylvania.
Bunge, M., and A., Carboni (1995). The symmetric topos. J. Pure Appl. Algebra 105: 233–249.
Carboni, A., and R., Celia Magno (1982). The free exact category on a left exact one. J. Austral. Math. Soc. Ser. A 33: 295–301.
Carboni, A., and E. M., Vitale (1998). Regular and exact completions. J. Pure Appl. Algebra 125: 79–116.
Centazzo, C., J., Rosický, and E. M., Vitale (2004). A characterization of locally D-presentable categories. Cah. Top. Géom. Différ. Catég. 45: 141–146.
Centazzo, C., and E. M., Vitale (2002). A duality relative to a limit doctrine. Theory Appl. Categ. 10: 486–497.
Cohn, P. M. (1965). Universal Algebra. Harper and Row.
Diers, Y. (1976). Type de densité d'une sous-catégorie pleine. Ann. Soc. Sc. Bruxelles 90: 25–47.
Dukarm, J. J. (1988). Morita equivalence of algebraic theories. Colloq. Math. 55: 11–17.
Ehresmann, C. (1963). Catégories structurées. Ann. Sci. École Norm. Sup. 80: 349–426.
Ehresmann, C. (1967). Sur les structures algébriques. C. R. Acad. Sci. Paris 264: A840–A843.
Eilenberg, S. (1961). Abstract description of some basic functors. J. Indian Math. Soc. (N. S.) 24: 231–234.
Elkins, B., and J. A., Zilber (1976). Categories of actions and Morita equivalence. Rocky Mountain J. Math. 6: 199–225.
Gabriel, P. (1962). Des catégories abéliennes. Bull. Sci. Math. France 90: 323–448.
Gabriel, P., and F., Ulmer (1971). Lokal Präsentierbare Kategorien. Lect. Notes in Math. 221. Springer.
Gran, M., and E. M., Vitale (1998). On the exact completion of the homotopy category. Cahiers Top. Géom. Diff. Cat. 39: 287–297.
Grätzer, G. (2008). Universal Algebra. Springer.
Hagemann, J., and C., Hermann (1979). A concrete ideal multiplication for algebraic systems and its relation to congruence modularity. Arch. Math. (Basel) 32: 234–245.
Heller, A. (1965). MR0163940. Math. Rev. 29.
Higgins, P. J. (1963–1964). Algebras with a scheme of operators. Math. Nachr. 27: 115–132.
Hyland, M., and J., Power (2007). The category theoretic understanding of universal algebra: Lawvere theories and monads. Electron. Notes Theor. Comput. Sci. 172: 437–458.
Janelidze, Z. (2006). Closedness properties of internal relations I: A unified approach to Mal'tsev, unital and subtractive categories. Theory Appl. Categ. 16: 236–261.
Janelidze, G., and M. C., Pedicchio (2001). Pseudogroupoids and commutators. Theory Appl. Categ. 8: 408–456.
Joyal, A. (2008). Notes on logoi (unpublished book in preparation).
Kelly, G. M. (1982). Basic Concepts of Enriched Categories. Cambridge University Press.
Lack, S., and J., Rosický (2010). Notions of Lawvere theory. Appl. Cat. Struct. (In press).
Lair, C. (1996). Sur le genre d'esquissabilité des catégories modelables (accessibles) possédant les produits de deux. Diagrammes 35: 25–52.
Lambek, J. (1968). A fixpoint theorem for complete categories. Math. Z. 103: 151–161.
Lawvere, F. W. (1963). Functorial semantics of algebraic theories. Dissertation, Columbia University.
Lawvere, F. W. (1969). Ordinal sums and equational doctrines. In Seminar on Triples and Categorical Homology Theory. Lect. Notes in Math. 80: Springer, 141–155.
Lawvere, F. W. (2008). Core varieties, extensivity, and rig geometry. Theory Appl. Categ. 20: 497–503.
Linton, F. E. J. (1966). Some aspects of equational categories. In Proc. Conf. Categorical Algebra La Jolla 1965. Springer, 84–94.
Linton, F. E. J. (1969a). Coequalizers in categories of algebras. In Seminar on Triples and Categorical Homology Theory. Lect. Notes in Math. 80. Springer, 75–90.
Linton, F. E. J. (1969b). Applied functorial semantics, II. In Seminar on Triples and Categorical Homology Theory. Lect. Notes in Math. 80. Springer, 53–74.
Loday, J.-L. (2008). Generalized Bialgebras and Triples of Operads. Astérisque.
Lurie, J. (2009). Higher Topos Theory. Princeton University Press.
Mac Lane, S. (1963). Natural associativity and commutativity. Rice Univ. Stud. 49: 38–46.
Mac Lane, S. (1998). Categories for the Working Mathematician. Springer.
Makkai, M., and R., Paré (1989). Accessible Categories: The Foundation of Categorical Model Theory. Contemp. Math. 104. AMS.
Manes, E. G. (1976). Algebraic Theories. Springer.
McKenzie, R. N. (1996). An algebraic version of categorical equivalence for varieties and more general categories. Lect. Notes Pure Appl. Mathematics 180. Marcel Dekker, 211–243.
Mitchell, B. (1965). Theory of Categories. Academic.
Morita, K. (1958). Duality for modules and its applications to the theory of rings with minimum conditions. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6: 83–142.
Pareigis, B. (1970). Categories and Functors. Academic.
Pedicchio, M. C. (1995). A categorical approach to commutator theory. J. Algebra 177: 647–657.
Pedicchio, M. C., and J., Rosický (1999). Comparing coequalizer and exact completions. Theory Appl. Categ. 6: 77–82.
Pedicchio, M. C., and F., Rovatti (2004). Algebraic categories. In Categorical Foundations. Cambridge University Press, 269–310.
Pedicchio, M. C., and E. M., Vitale (2000). On the abstract characterization of quasivarieties. Algebra Universalis 43: 269–278.
Pedicchio, M. C., and R. J., Wood (2000). A simple characterization of theories of varieties. J. Algebra 233: 483–501.
Popesco, N., and P., Gabriel (1964). Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes. C. R. Acad. Sci. Paris 258: 4188–4190.
Porst, H.-E. (2000). Equivalence for varieties in general and for Bool in particular. Algebra Universalis 43: 157–186.
Power, J. (1999). Enriched Lawvere theories. Theory Appl. Categ. 6: 83–93.
Rezk, C. W. (1996). Spaces of algebra structures and cohomology of operands. Thesis, MIT.
Roos, J. E. (1965). Caractérisation des catégories qui sont quotients des modules par des sous-catégories bilocalisantes. C. R. Acad. Sci. Paris 261: 4954–4957.
Rosický, J. (2007). On homotopy varieties. Adv. Math. 214: 525–550.
Rosický, J., and E. M., Vitale (2001). Exact completion and representations in abelian categories. Homology Homotopy Appl. 3: 453–466.
Schubert, H. (1972). Categories. Springer.
Street, R. (1974). Elementary cosmoi I. In Category Seminar Sydney. Lect. Notes in Math. 420. Springer, 75–103.
Street, R., and R. F. C., Walters (1978). Yoneda structures on 2-categories. J. Algebra 50: 350–379.
Tholen, W. (2003). Variations on the Nullstellensatz, paper presented at the European Conference on Category Theory, Haute Bodeux, Belgium.
Ulmer, F. (1968). Properties of dense and relative adjoints. J. Algebra 8: 77–95.
Vitale, E. M. (1996). Localizations of algebraic categories. J. Pure Appl. Algebra 108: 315–320.
Vitale, E. M. (1998). Localizations of algebraic categories II. J. Pure Appl. Algebra 133: 317–326.
Voronov, A. A. (2005). Notes on universal algebra. In Proc. Symp. Pure Math 73. AMS, 81–103.
Watts, C. E. (1960). Intrinsic characterizations of some additive functors. Proc. Amer. Math. Soc. 11: 5–8.
Wechler, W. (1992). Universal Algebra for Computer Science. Springer.
Wraith, G. C. (1970). Algebraic Theories. Aarhus Lecture Note Series 22. Aarhus Universitat.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.