On descent for coalgebras and type transformations

Document Type: Research Paper


Laboratory of Algebra, Geometry and Applications, Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O. Box 812, Yaounde, Republic of Cameroon.


We find a criterion for a morphism of coalgebras over a Barr-exact category to be effective descent and determine (effective) descent morphisms for coalgebras over toposes in some cases. Also, we study some exactness properties of endofunctors of arbitrary categories in connection with natural transformations between them as well as those of functors that these transformations induce between corresponding categories of coalgebras.  As a result, we find conditions under which the induced functors preserve natural number objects as well as a criterion for them to be exact. Also this enable us to give a criterion for split epis in a category of coalgebras to be effective descent.


[1] J. Adamek, Introduction to coalgebra, Theory Appl. Categ. 14(8) (2005), 157-199.
[2] J. Adamek, H.P. Gumm, and V. Trnkova, Presentation of set functors: A coalgebraic
perspective, preprint.
[3] J. Adamek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories",
John Wiley and Sons, New York, 1990.
[4] J. Adamek and H.E. Porst, On varieties and covarieties in a category, Math. Structures
Comput. Sci. 13(2) 2003, 201-232.
[5] M. Barr, Right exact functors, J. Pure Appl. Algebra 4 (1974), 1-8.
[6] M. Barr and C.Wells, Toposes, Triples and Theories, Theory Appl. Categ. 12 (2005),
[7] M.M. Clementino and G. Janelidze, A note on e ective descent morphisms of topo-
logical spaces and relational Algebras, Topology Appl. 158(17), (2011) 2431-2436.

[8] T. Everaert, E ective descent morphisms of regular epimorphisms, J. Pure Appl.
Algebra 216 (2012), 1896-1904.
[9] M. Gran, Notes on regular, exact and additive categories", Summer School on Category
Theory and Algebraic Topology, Ecole Polythecnique Federale de Lausanne,
11-13 September 2014.
[10] H.P. Gumm, From T-coalgebras to lter structures and transition systems, Chapter:
Algebra and Coalgebra in Computer Science, Lecture Notes in Comput. Sci. 3629
(2005), 194-212.
[11] H.P. Gumm, On coalgebras and type transformations, Discuss. Math. Gen. Algebra
Appl. 27 (2007), 187-197.
[12] H.P. Gumm, Elements of the general theory of coalgebras", LUATCS'99, Rand
Africaans University, Johannesburg, South Africa, 1999.
[13] H.P. Gumm, On minimal coalgebras, Appl. Categ. Structures 16(3) (2008), 313-332.
[14] H.P. Gumm and T. Schroder, Coalgebraic structures from weak limit preserving
functors, Electron. Notes Theor. Comput. Sci. 33 (2000), 113-133.
[15] H.P. Gumm and T. Schroder, Types and coalgebraic structure, Algebra Universalis
53 (2005), 229-252.
[16] H.P. Gumm and T. Schroder, Coalgebras of bounded types, Math. Structures Comput.
Sci. 12 (2002), 565-578.
[17] H.P. Gumm and T. Schroder, Products of coalgebras, Algebra Universalis 46 (2001),
[18] G. Janelidze and W. Tholen, Facets of descent, I, Appl. Categ. Structures 2 (1994),
[19] G. Janelidze and W. Tholen, Facets of descent, III: Monadic descent for rings and
algebras, Appl. Categ. Structures 12 (2004), 461-477.
[20] G. Janelidze, M. Sobral, and W. Tholen, Beyond Barr exactness: e ective descent
morphisms, Chapter VIII: Categorical Foundations: Special Topics in Order, Topology,
Algebra and Sheaf Theory, Encyclopedia of Math. Appl. 97, Cambridge University
Press (2004), 359-405.
[21] P. Johnstone, J. Power, T. Tsujishita, H.Watanabe, and J.Worrell, On the structure
of categories of coalgebras, Theoret. Comput. Sci. 260 (2001), 87-117.
[22] A. Joyal and M. Tierney, An extension of Galois Theory of Grothendieck, Mem.
Amer. Math. Soc. 51 (309) (1984).

[23] M. Kianpi and C. Nkuimi-Jugnia, A simpli cation functor for coalgebras, Int. J.
Math. Math. Sci. 2006 (2006), 1-9.
[24] M. Kianpi and C. Nkuimi-Jugnia, On simple and extensional coalgebras beyond Set,
Arab. J. Sci. Eng. 33(2C) (2008), 295-313.
[25] M. Kianpi and C. Nkuimi-Jugnia, A note on descent for coalgebras, Homology Homotopy
Appl., to appear.
[26] B. Mesablishvili, Descent in categories of (co)algebras, Homology Homotopy Appl.
7(1) (2005), 1-8.
[27] S. Lack, An embedding theorem for adhesive categories, Theory Appl. categ. 25(7)
(2011), 180-188.
[28] S. Lack and P. Sobocinski, Toposes are adhesive, Chapter: Graph transformations,
Lecture Notes in Comput. Sci. 4178 (2006), 184-198.
[29] M. Makkai, A theorem on Barr-Exact categories, with in nitary generalization, Ann.
Pure Appl. Logic 47 (1990), 225-268.
[30] M. Menni, A characterization of the left exact categories whose exact completions
are toposes, J. Pure Appl. Algebra 177 (2003), 287-301.
[31] I. Moerdijk, Descent theory for toposes, Bull. Soc. Math. Belgique 41(2) (1989),
[32] J. Reiterman and W. Tholen, E ective descent maps of topological spaces, Topology
Appl. 57 (1994), 53-69.
[33] A.H. Roque, Notes on e ective descent and projectivity in quasivarieties of universal
algebras, Theory Appl. Categ. 21(9) (2008), 172-181.
[34] J.J.M.M. Rutten, Universal coalgebra: a theory of systems, Theoret. Comput. Sci.
249 (2000), 3-80.
[35] J.D.H. Smith, Permutation representations of left quasigroups, Algebra Universalis
55 (2006), 387-406.
[36] J. Worrell, A note on coalgebras and presheaves, Math. Structures Comput. Sci.
15(3) (2005), 475-483.