nLab descent morphism

Idea

Given a fibered category, a morphism along which the induced comparison functor between the category of descent data and the codomain fibration is fully faithful (resp. an equivalence of categories) is said to be a descent morphism (resp. effective descent morphism).

The case of codomain fibration

Definition

Let $C$ be a category with pullbacks and coequalizers. For any morphism $p\colon A\to B$, we have an internal category $ker(p)$ defined by $A\times_B A \rightrightarrows A$ (the kernel pair of $p$). The category of descent data for $p$ is the category $C^{ker(p)}$ (the “descent object”) of internal diagrams on this internal category. Explicitly, an object of $C^{ker(p)}$ is a morphism $C\to A$ together with an action $A\times_B C \to C$ satisfying suitable axioms.

The evident internal functor $ker(p) \to B$ (viewing $B$ as a discrete internal category) induces a comparison functor $C^B \to C^{ker(p)}$. We say that $p$ is:

• a descent morphism if this comparison functor is fully faithful, and
• an effective descent morphism if this comparison functor is an equivalence of categories.

It is a little unfortunate that the more important notion of effective descent has the longer name, but it seems unwise to try to change it (although the Elephant uses “pre-descent” and “descent”).

Properties

Let $C$ be a category with pullbacks.

Theorem

$p\colon A\to B$ is a descent morphism if and only if $p$ is a stable regular epimorphism.

In particular, descent morphisms are closed under pullback and composition. Moreover, in a regular category, the descent morphisms are precisely the regular epimorphisms.

Theorem

A split epimorphism is an effective descent morphism.

See (JT, Proposition 2.4),

Perhaps more surprising is:

Theorem

Effective descent morphisms are closed under pullback and composition.

Closure under composition was proved in (ST) under the additional assumption that $C$ has coequalisers, and in the case of just having finite limits, in (RST). Closure under pullback was proved in (JST).

General case

In general, descent is about higher sheaf conditions (i.e. stack conditions). More precisely, being an $n$-stack means that all covers in the base are effective $n$-categorical descent morphism. Hence the morphism being of effective descent is a building block, the single morphism case of a stack condition.

Thus, being an effective descent morphism says that the corresponding fibered category is a 1-stack (“2-sheaf”) for the singleton covering family $p$. Similarly, $p$ is a descent morphism iff the codomain fibration is a pre-stack (that is, a 2-separated 2-presheaf) for $p$.

More generally, we may use the terms “descent morphism” and “effective descent morphism” relativized to any fibration or indexed category rather than the codomain fibration.

We can also, of course, generalize to higher categories: an n-category with pullbacks has an analogue of a “codomain fibration”, and we can ask for stack conditions on it. This is most common in the case of (infinity,1)-categories; see the page descent for more information and links.

Descent can sometimes (for this we need to have also the direct image functor) be rephrased in terms of the monadicity theorem; see monadic descent.

Examples

If $C$ is exact, or has stable reflexive coequalizers, then every regular epimorphism is an effective descent morphism. (See, for instance, section B1.5 of the Elephant.) In particular, this is the case for any topos.

However, there are also important effective descent morphisms in non-exact categories.

• In Top, there are characterizations of effective descent morphisms, see CJ20.

• In the category Loc of locales, every triquotient map? is an effective descent morphism. These includes open surjections and also proper surjections.

…

Of course, there are also many effective descent morphisms relative to fibrations other than the codomain fibration. If $A$ is a stack for a particular Grothendieck topology, then every singleton cover in that topology will be, by definition, an effective descent morphism relative to $A$. A few important examples are:

• Every fpqc cover of schemes is an effective descent morphism relative to the indexed category $QCoh(-)$ of quasicoherent sheaves.

…

Related concepts

• descent

  • cover

  • cohomological descent

  • descent morphism

  • monadic descent,

    • Sweedler coring

    • higher monadic descent

    • descent in noncommutative algebraic geometry

• sheaf, (2,1)-sheaf, 2-sheaf (∞,1)-sheaf

References

Original articles:

• Alexander Grothendieck, Def. 1.7 in: Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire N. Bourbaki exp. no190 (1960) 299–327 (Numdam) (part of FGA)

• SGA I.6 Catégories fibrées et descente (exposé VI by Pierre Gabriel; exp. VIII, exp. IX) in: A. Grothendieck, M. Raynaud, SGA I

• Jean Giraud, Méthode de la descente, Mémoires de la S. M. F., tome 2 (1964) 156 pp. (Numdam)

See also:

• G. Janelidze, W. Tholen. How algebraic is the change-of-base functor?, Category Theory: Proceedings of the International Conference held in Como, Italy, July 22–28, 1990. pp 174–186 Lecture Notes in Mathematics 1488 Springer 1991. doi:10.1007/BFb0084219.

• M. Sobral, W. Tholen, Effective descent morphisms and effective equivalence relations, Category Theory 1991, CMS Conference Proceedings 13 (1992) pp 421–433.

• J. Reiterman, M. Sobral, W. Tholen, Composites of effective descent maps, Cahiers 34 (1993) pp 193–207, numdam.

• George Janelidze, Manuela Sobral and Walter Tholen, Beyond Barr Exactness: Effective Descent Morphisms, in: Categorical Foundations Special Topics in Order, Topology, Algebra, and Sheaf Theory, pp. 359–406 Cambridge University Press 2003, doi:10.1017/CBO9781107340985.011.

• Maria Manuel Clementino, George Janelidze, Another note on effective descent morphisms of topological spaces and relational algebras, Topology Appl. 273 (2020), pg 106961, doi:10.1016/j.topol.2019.106961.

A criterion for categories (including quasi-abelian categories) under which effective descent morphisms and descent morphisms coincide is established in

• Marino Gran, Olivette Ngaha Ngaha, Effective descent morphisms in star-regular categories, Homology, Homotopy and Applications 15 Number 2 (2013) pp 127–144, doi:10.4310/hha.2013.v15.n2.a7.