In its more specific meaning descent is the study of generalizations of the sheaf condition on presheaves to presheaves with values in higher categories. Those higher presheaves that satisfy descent are called infinity-stacks.
More generally, descent theory studies existence and (non)uniqueness of an object $u$ in a (possibly higher) category $C_X$ provided some “inverse image” functor $f^*:C_X\to C_Y$ which applied to $u$ produces an object in some (possibly higher) category $C_Y$ (or a collection of inverse image functors $f^*_\alpha(u)$ in $D_\alpha$ is given). In favourable cases, the nonuniqueness is parametrized by equipping the object $f^*(u)$ with additional “gluing” data $\xi$. The pair $(f^*(u),\xi)$ is called a descent datum, the existence of a reconstruction procedure of $u$ from $(f^*(u),\xi)$ is also called a descent, and it describes the property that the (higher) category of descent data in $C_Y$ is equivalent to the category $C_X$, or at least that it embeds via a canonical fully faithful functor.
The most important case is when there is a descent (in the sense of equivalence of higher categories) along an inverse image functor along every cover of a Grothendieck topology or its higher analogue; though many cases (for example descent in noncommutative algebraic geometry) do not fit into this framework. These cases of descent along all covers is also called (higher) stack theory and may be phrased in modern viewpoint as a characterization of $(\infty,1)$-sheaves (i.e. $\infty$-stacks) among all $(\infty,1)$-presheaves as those $(\infty,1)$-presheaves which are local objects with respect to certain morphisms $Y \to X$ which are to be regarded as covers or hypercover of the $(\infty,1)$-presheaf $X$: the idea is that an $(\infty,1)$-sheaf “descends from the cover $Y$ down to $X$”.
More concretely
every (∞,1)-category of (∞,1)-sheaves is characterized as being a sub-(∞,1)-topos $Sh(S) \hookrightarrow PSh(S)$ of the $(\infty,1)$-topos of (∞,1)-presheaves on some (small) (∞,1)-category $S$;
every such $(\infty,1)$-topos is a reflective (∞,1)-subcategory of $PSh(S)$, hence a localization of an (∞,1)-category at a collection $W = \{Y \to X\}$ of morphisms which are sent to equivalences by the left adjoint of the inclusion;
and the sheaves in $Sh(S) \hookrightarrow PSh(S)$ are precisely the local objects with respect to this collection $W$ of morphisms, i.e. precisely those objects $A \in PSh(S)$ such that $PSh(S)(X,A) \to PSh(S)(Y,A)$ is an isomorphism in the homotopy category, which we shall write $PSh(S)(X,A) \stackrel{\simeq}{\to} PSh(S)(Y,A)$ in the following paragraphs.
This condition is essentially the descent conditon.
In concrete models for the (∞,1)-category of (∞,1)-sheaves – notably in terms of the model structure on simplicial presheaves – the morphisms $Y \to X$ in $W$ usually come from hypercovers $Y \to X$;
in this case the above condition becomes $PSh(S)(X,A) \stackrel{\simeq}{\to} PSh(S)(colim^\Delta Y_\bullet, A)$ which is equivalent to $PSh(S)(X,A) \stackrel{\simeq}{\to} lim^\Delta PSh(S)(Y_\bullet, A)$. This in turn is usually equivalently written
And this is the form of the local object-condition which is usually called descent condition.
Descent is best understood as a direct generalization of the situation for 0-stacks, i.e. ordinary sheaves, which we briefly recall in a language suitable for the following generalization.
For $S$ any small category and Set the category of small sets, write $\mathrm{PSh}(S) = [S^{op}, Set]$ for the category of presheaves on $S$. Categories of this form enjoy various nice properties which are familiar from $Set$ itself, and which are summarized by saying that $\mathrm{PSh}(S)$ is a topos. The relevance of this for the present purpose is that there is a natural notion of morphisms of topoi, which are functors respecting this structure in some sense: these are called geometric morphisms.
A category of sheaves on $S$ is a sub-topos of $PSh(S)$ in that it is a full and faithful functor $Sh(S)\hookrightarrow PSh(S)$ which is a geometric morphism.
One finds that the reflective subcategory $Sh(S) \hookrightarrow PSh(S)$ of sheaves inside presheaves is the localization of $PSh(S)$ at morphisms $f : Y \to X$ called local isomorphisms, which are determined by and determing the choice of topos-inclusion. A presheaf $A$ is a sheaf precisely if it is a local object with respect to these local isomorphisms, that is precisely if
is an isomorphism for all local isomorphisms $f$.
This locality condition is in fact the descent condition: the sheaf has to descend from $Y$ down to $X$. More concretely, this condition is called a descent condition when evaluated on morphisms $f : Y \to X$ which are hypercovers:
namely if $\pi : Y^1 \to X$ is a local epimorphism with respect to the coverage that corresponds to the localization and if $\pi_2 : Y^2 \to Y^1 \times_X Y^1$ is a local epimorphism, then with
being the two canonical morphisms out of $Y^2$, it follows that the canonical morphism
is a local isomorphism.
(This is excercise 16.6 in Categories and Sheaves).
Therefore for a presheaf $A$ to be a sheaf, it is necessary that
is an isomorphism. The colimit may be taken out of the hom-functor to make this equivalently
It is convenient, suggestive and common to write $A(X) := Hom_{PSh(S)}(X,A)$, $A(Y^\bullet) := Hom_{PSh(S)}(Y^\bullet,A)$, following the spirit of the Yoneda lemma whether or not $X$ and/or $Y^\bullet$ are representable. In that notation the above finally becomes
This is the form of the condition that is most commonly called the descent condition.
For more references and background on the following see descent for simplicial presheaves.
A well-studied class of models/presentations for an (∞,1)-category of (∞,1)-sheaves is obtained using the model structure on simplicial presheaves on an ordinary (1-categorical) site $S$, as follows.
Let $[S^{op}, SSet]$ be the SSet-enriched category of simplicial presheaves on $S$.
Recall from model structure on simplicial presheaves that there is the global and the local injective simplicial model structure on $[S^{op}, SSet]$ which makes it a simplicial model category and that the local model structure is a (Bousfield-)localization of the global model structure.
So in terms of simplicial presheaves the localization of an (∞,1)-category that we want to describe, namely ∞-stackification, is modeled as the localization of a simplicial model category.
Recall that the (∞,1)-category modeled/presented by a simplicial model category is the full SSet-subcategory on fibrant-cofibrant objects. According to section 6.5.2 of HTT we have:
the full simplicial subcategory on fibrant-cofibrant objects of $[S^{op}, SSet]$ with respect to the global injective model structure is (the SSet-enriched category realization of) the $(\infty,1)$-category $PSh_{(\infty,1)}(S)$ of (∞,1)-presheaves on $S$.
the full simplicial subcategory on fibrant-cofibrant objects of $[S^{op}, SSet]$ with respect to the local injective model structure is (the SSet-enriched category realization of) the $(\infty,1)$-category $\bar{Sh}_{(\infty,1)}(S)$ which is the hypercompletion of the $(\infty,1)$-category $Sh_{(\infty,1)}(S)$ of (∞,1)-sheaves on $S$.
Since with respect to the local or global injective model structure all objects are automatically cofibrant, this means that $\bar Sh_{(\infty,1)}(S)$ is the full sub-$(\infty,1)$-category of $PSh_{(\infty,1)}(S)$ on simplicial presheaves which are fibrant with respect to the local injective model structure: these are the ∞-stacks in this model.
By the general properties of localization of an (∞,1)-category there should be a class of morphisms $f : Y \to X$ in $PSh_{(\infty,1)}(S)$ – hence between injective-fibrant objects in $[S^{op}, PSh(S)]$ – such that the simplicial presheaves representing $\infty$-stacks are precisely the local objects with respect to these morphisms.
The general idea of descent in this simplicial context is the precise analog of the situation for ordinary sheaves, but with ordinary (co)limits replaced everywhere with the (∞,1)-categorical (co)limits, which in terms of the presentation by the model structure on simplicial presheaves amounts to the homotopy (co)limit.
So for $Y \to X$ a morphism of simplicial presheaves, the condition that a simplicial presheaf $A$ is local with respect to it, hence satisfies descent with respect to it, is generally that
is a weak equivalence, where $RHom$ denotes the corresponding $(\infty,1)$-categorical hom, i.e. the derived hom with respect to the model structure on simplicial presheaves – for instance the ordinary simplicial hom if both $Y$ and $A$ are fibrant with respect to the given model structure.
The details on which morphisms $Y \to X$ one needs to check against here have been worked out in
We now describe central results of that article.
For $X \in S$ an object in the site regarded as a simplicial presheaf and $Y \in [S^{op}, SSet]$ a simplicial presheaf on $S$, a morphism $Y \to X$ is a hypercover if it is a local acyclic fibration, i.e. of for all $V \in S$ and all diagrams
there exists a covering sieve $\{U_i \to V\}$ of $V$ with respect to the given Grothendieck topology on $S$ such that for every $U_i \to V$ in that sieve the pullback of the abve diagram to $U$ has a lift
If $S$ is a Verdier site then every such hypercover $Y \to X$ has a refinement by a hypercover which is cofibrant with respect to the projective global model structure on simplicial presheaves. We shall from now on make the assumption that the hypercovers $Y \to X$ we discuss are cofibrant in this sense. These are called split hypercovers. (This works in many cases that arise in practice, see the discussion after DHI, def. 9.1.)
The objects of $Sh_{(\infty,1)}(S)$ – i.e. the fibrant objects with respect to the projective model structure on $[S^{op}, SSet]$ – are precisely those objects $A$ of $PSh_{(\infty,1)}(S)$ – i.e. Kan complex-valued simplicial presheaves – which satisfy descent for all split hypercovers, i.e. those for which for all split hypercover $f : Y \to X$ in $[S^{op}, SSet]$ we have that
This is DHI, thm 1.3 formulated in the light of DHI, lemma 4.4 ii).
Notice that by the co-Yoneda lemma every simplicial presheaf $F : S^{op} \to SSet$, which we may regard as a presheaf $F : \Delta^{op}\times S^{op} \to Set$, is isomorphic to the weighted colimit
which is equivalently the coend
where $F_n$ is the Set-valued presheaf of $n$-cells of $F$ regarded as an $SSet$-valued presheaf under the inclusion $Set \hookrightarrow SSet$, and where the SSet-weight is the canonical cosimplicial simplicial set $\Delta$, i.e. for all $X \in S$
In particular therefore for $A$ a Kan complex-valued presheaf the descent condition reads
With the shorthand notation introduced above the descent condition finally reads, for all global-injective fibrant simplicial presheaves $A$ and hypercovers $U \to X$:
The right hand here is often denoted $Desc(Y_\bullet \to X, A)$, in which case this reads
While simplicial sets are a very convenient model for general reasoning about higher weak categories and ∞-groupoids, often concrete computations in particular with $(\infty)$-groupoids are more convenient in the context of more strictified models.
Notably, by the generalized Dold-Kan correspondence the ω nerve injects crossed complexes – nonabelian generalizations of chain complexes of abelian groups which are equivalent to strict ω-groupoids – to simplicial sets
Since for instance something as simple as an abelian group $A$ regarded as a complex of groups in degree $n$ (hence as an $n$-group) already bcomes a somewhat involved object to understand under the nerve operation,
it is desireable to have a means to control descent for simplicial presheaves which happen to factor through the $\omega$-nerve directly in the context of $Str \omega Cat$.
In his work on descent
Ross Street considered presheaves with values in strict ω-categories
and declared the descent $\omega$-category with respect to a simplicial object $Y_\bullet : \Delta^{op} \to S$ to be the weighted limit in $Str\omega-Cat$-enriched category theory
where $O := F \Delta : \Delta \to Str \omega Cat$ are the orientals, i.e. the free $\omega$-categories on the simplicial simplices
where $F : SSet \to Str\omega Cat$ is the right adjoint to the ω-nerve $N : Str \omega Cat \to SSet$.
The two precscriptions
have a very similar appearance. The following theorem asserts if and when they are actually equivalent.
There exists a canonical comparison map
This is a weak equivalence of Kan complexes if the cosimplicial simplicial set $N(A(U^\bullet))$ is Reedy fibrant.
The full proof is given at Verity on descent for strict omega-groupoid valued presheaves.
We unwrap the expression
for the descent data for a presheaf $A$ with respect to a (hyper)cover $Y \to X$
This weighted limit (whether taken in $SSet$- or in $Str \omega Cat$-enriched category theory) is given by the coend
Unwrapping what this means one finds that an object/vertex of this is a choice of $n$-simplex in each $A(Y_n)$, subject to conditions which say that the boundary of this $n$-simplex must be obtained from pullback of $A$ along the maps $Y_n \to Y_{[n-1]}$ of the $(n-1)$-simplex in $A(Y_{n-1})$
Namely an object in
$lim^W\Delta A(Y_\bullet) \simeq \int^{[n] \in \Delta} [\Delta^n, A(Y_n)]$
is a commuting diagram
where the vertical arrows indicate all the simplicial maps of the cosimplicial objects $\Delta$ and $A(Y_\bullet)$.
So this is
on $Y_0$ an object $a \in A(Y_0)$;
on “double intersections” (might be a cover of double intersections) $Y_1$ a gluing isomorphism $g : \pi_1^* a \to \pi_2^* a$ which identifies the two copies of $a$ obtained by pullback along the two projection maps $\pi_1, \pi_2 : U \times_X U \to U$.
on “triple intersections” $Y_2$ a gluing 2-isomorphism $\array{ && \pi_2^* a \\ & {}^{\pi_{12}^* g}\nearrow &\Downarrow^f& \searrow^{\pi_{23}^* g} \\ \pi_1^* a && \stackrel{\pi_{13}^* g}{\to} && \pi_3^* a }$ which identifies the different gluing 1-isomorphisms.
And so on.
The article
spells out how the familiar formulation of the descent condition for ordinary stacks is equivalent to the corresponding descent condition for simplicial presheaves, duscussed above.
Sometimes one wishes to compute the descent objects for presheaves of the form
where $B : S \to [S^{op}, SSet]$ is a given presheaf-valued co-presheaf. For instance in the context of differential nonabelian cohomology one is interested in the co-presheaf that assigns fundamental ∞-groupoids
in which case the presheaf
would assign to $U \in S$ the pre-$\infty$-stack of “trivial $A$-principial bundles with flat connection”.
For $Y \to X$ a given (hyper)cover, the descent object for $[B(-), A]$ can be expressed as
This way the descent for $[B(-),A]$ on the object $U = colim^\Delta U_\bullet$ is reexpressed as descent for $A$ of the $B$-modified object $colim^\Delta B(Y_\bullet)$. Following Street, this we may call the codescent object, as it co-represents descent.
In some context the descent condion may algebraically be encoded in an adjunction. This leads to the notion of monadic descent. See there for more details.
descent