Descent theory is the study of local-to-global problems. After a quick review of simple definitions of category theory, we give some algebraic and topological examples of such problems. Using no more than basic topological tools, we motivate and state a descent-theoretical problem, whose solution was found in 1994 by Reiterman and Tholen, [3]. We proceed to explain how to generalize this example, and discuss its applications in algebra and topology. This talk is based on [2] and [1].

[1] G. Janelidze, M. Sobral, W. Tholen, Beyond Barr Exactness: Effective Descent Morphisms, In M. Pedicchio & W. Tholen (Eds.), Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory, pp. 359-406.

[2]  G. Janelidze, W. Tholen, Facets of Descent I, Appl. Categorical Structures 2 (1994) 245-281.

[3]  J. Reiterman, W. Tholen, Effective descent maps of topological spaces, Topology Appl. 57 (1994) 53-69.