Given a fibered category, a morphism along which the induced comparison functor between the category of the descent data and the codomain fiber is fully faithful (equivalence of categories) is said to be a descent morphism (resp. effective descent morphism).
Let $C$ be a category with pullbacks. 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:
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”).
Let $C$ be a category with pullbacks.
$p\colon A\to B$ 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.
Perhaps more surprising is:
Effective descent morphisms are closed under pullback and composition.
See (ST) and (RST) for proofs.
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.
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 is a characterization…
In the category Loc of locales, every triquotient map? is an effective descent morphism. These includes open surjections and also proper surjections.
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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:
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