Towards free localic algebras

Document Type : Research Paper

Authors

Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South Africa. National Institute for Theoretical and Computational Sciences (NITheCS), South Africa.

10.48308/cgasa.2024.233933.1445

Abstract

The purpose of this paper is to establish that the underlying object
functor from the models of a Lawvere theory to the base category creates limits
and coequalisers of all parallel pairs of homomorphisms whose underlying pair
admit a split coequaliser. We show that for a small complete category with
a well behaved proper factorisation structure, the underlying functor admits
a left adjoint and the category of such models is precisely monadic over the
base category in the sense of Beck’s Theorem. In particular, this establishes
the existence of free localic algebras for any Lawvere theory, generalising the
known results for the existence of free localic groups.

Keywords

Main Subjects

References

[1] Ad´amek, J., Herrlich, H., and Strecker, G.E., “Abstract and Concrete Categories: the Joy of Cats”, Repr. Theory Appl. Categ. 17, 1-507, 2006.
[2] Ad´amek, J. and Rosick´y, J., “Locally Presentable and Accessible Categories”, Lond. Math. Soc. Lect. Note Ser. 189, Cambridge University Press, 1994.
[3] Ad´amek, J., Rosick´y, J., and Vitale, E.M., “Algebraic Theories: A Categorical Introduction to General Algebra with a foreword by F.W. Lawvere”, Camb. Tracts Math. 184, Cambridge University Press, 2011.
[4] Bastiani, A. and Ehresmann, C., Categories of sketched structures, Cah. Topologie G´eom. Diff´er. Cat´egoriques 13 (1972), 105-214.
[5] B´enabou, J., Structures alg´ebriques dans les cat´egories, Cah. Topologie G´eom. Diff´er. Cat´egoriques 10 (1968), 1-126.
[6] Birkhoff, G., On the structure of abstract algebras, Proc. Camb. Philos. Soc. 31 (1935), 433-454.
[7] Birkhoff, G. and Lipson, J.D., Heterogeneous algebras, J. Comb. Theory 8 (1970), 115-133.
[8] Borceux, F., “Handbook of Categorical Algebra, Volume 1: Basic Category Theory”, Encycl. Math. Appl. 50, Cambridge University Press, 2008.
[9] Borceux, F. and Bourn, D., “Mal’cev, Protomodular, Homological and Semi-Abelian Categories”, Mathematics and its Applications 566, Springer, 2004.
[10] Borceux, F., and Clementino, M.M., Topological semi-abelian algebras, Adv. Math. 190(2) (2005), 425-453.
[11] Borceux, F., and Clementino, M.M., Topological protomodular algebras, Topology Appl. 153(16) (2006), 3085-3100.
[12] Bourn, D., and Janelidze, G., Characterization of protomodular varieties of universal algebras, Theory Appl. Categ. 11 (2003), 143-147.
[13] Carboni, A., Janelidze, G., Kelly, G.M., and Par´e, R., On localization and stabilization for factorization systems, Appl. Categ. Struct. 5(1) (1997), 1-58.
[14] Clementino, M.M., “An Invitation to Topological Semi-Abelian Algebras”, in: New Perspectives in Algebra, Topology and Categories, Coimbra Mathematical Texts 1, Springer, 2021.
[15] Ehresmann, C., Sur les structures alg´ebriques, C.R. Acad. Sci., Paris, S´er. A 264 (1967), 840-843.
[16] Freyd, P.J. and Kelly, G.M., Categories of continuous functors. I, J. Pure Appl. Algebra 2 (1972), 169-191.
[17] Ghosh, P.P., Internal neighbourhood structures, Algebra Univers. 81(12) (2020), https://doi.org/10.1007/s00012-020-0640-2.
[18] Ghosh, P.P., Internal neighbourhood structures. II: Closure and closed morphisms, Categ. Gen. Algebr. Struct. Appl. 18(1) (2023), 155-223.
[19] Higgins, Ph.J., Algebras with a scheme of operators, Math. Nachr. 27 (1963), 115-132.
[20] Isbell, J., Kˇr´ıˇz, I., Pultr, A., and Rosick´y, J., Remarks on localic groups, in: Categorical Algebra and its Applications, Proc. 1st Conf., Louvain-la-Neuve/Belg. 1987, Lect. Notes Math. 1348 (1988), 154-172.
[21] Isbell, J.R., “Subobjects, Adequacy, Completeness and Categories of Algebras”, Diss. Math. 36, Instytut Matematyczny Polskiej Akademi Nauk, 1964.
[22] Lawvere, F.W., Functorial semantics of algebraic theories, Proc. Natl. Acad. Sci. USA 50 (1963), 869-872.
[23] Mac Lane, S., “Categories for the Working Mathematician”, Grad. Texts Math. 5, Springer, 1998.
[24] Monk, J.D., “Introduction to Set Theory”, McGraw-Hill Book Co., 1969.
[25] Picado, J. and Pultr, A., “Frames and Locales: Topology without Points”, Front. Math., Springer, 2012.
Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 26 August 2024

Files

Share

How to cite

Statistics