Descent (category theory)

In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. The reason for abstraction here is, at a fundamental level, that passage to a quotient space is not very well-behaved in topology: more accurately, it is a tribute to the efforts to use category theory to get round the alleged 'brutality' of imposing equivalence relations within geometric categories.

Related Topics:
Mathematics - Topology - Equivalence relation - Topological space - Quotient space - Category theory

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The ideas here flourished in the period 1955-1965 (which was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). From the point of view of category theory the work of comonads of Beck was a summation of those ideas. The difficulties of algebraic geometry with passage to the quotient are acute: it is like doing the non-commutative geometry of Connes, to mention the currently-fashionable theory in the area of 'bad quotients', but with polynomials to separate points, rather than general continuous functions. The urgency (to put it that way) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular. As with a number of the more abstract flights of the Grothendieck school, later work relied on some of this and bypassed other parts (to the extent that the papers, published only in mimeographed form, may have already become hard to find). The work a few years later of David Mumford in his geometric invariant theory spectacularly mixed scheme and categorical techniques with more concrete geometry, to construct moduli spaces for curves and abelian varieties (for the first time, in the required technical sense of 'moduli').

Related Topics:
Algebraic topology - Algebraic geometry - Category theory - Comonad - Non-commutative geometry - Grothendieck - Representable functor - Moduli problem - David Mumford - Geometric invariant theory - Scheme - Moduli space

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