Document Type : Research Paper

**Author**

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

**Abstract**

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.

**Keywords**

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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.

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53 (2005), 229-252.

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Sci. 12 (2002), 565-578.

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163-185.

[18] G. Janelidze and W. Tholen, Facets of descent, I, Appl. Categ. Structures 2 (1994),

245-281.

[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: eective 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.

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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),

373-391.

[32] J. Reiterman and W. Tholen, Eective descent maps of topological spaces, Topology

Appl. 57 (1994), 53-69.

[33] A.H. Roque, Notes on eective 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.

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 innitary 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),

373-391.

[32] J. Reiterman and W. Tholen, Eective descent maps of topological spaces, Topology

Appl. 57 (1994), 53-69.

[33] A.H. Roque, Notes on eective 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.