In its categorical 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 uu in a (possibly higher) category C XC_X provided some “inverse image” functor f *:C XC Yf^*:C_X\to C_Y which applied to uu produces an object in some (possibly higher) category C YC_Y (or a collection of inverse image functors f α *(u)f^*_\alpha(u) in D αD_\alpha is given). In favourable cases, the nonuniqueness is parametrized by equipping the object f *(u)f^*(u) with additional “gluing” data ξ\xi. The pair (f *(u),ξ)(f^*(u),\xi) is called a descent datum, the existence of a reconstruction procedure of uu from (f *(u),ξ)(f^*(u),\xi) is also called a descent, and it describes the property that the (higher) category of descent data in C YC_Y is equivalent to the category C XC_X, or at least that it embeds via a canonical fully faithful functor. Descent theory in 1-categorical context has been first formulated by Grothendieck in FGA using pseudofunctors and in SGA1 using fibered categories.

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 (,1)(\infty,1)-sheaves (i.e. \infty-stacks) among all (,1)(\infty,1)-presheaves as those (,1)(\infty,1)-presheaves which are local objects with respect to certain morphisms YXY \to X which are to be regarded as covers or hypercover of the (,1)(\infty,1)-presheaf XX: the idea is that an (,1)(\infty,1)-sheaf “descends from the cover YY down to XX”.

More concretely

  • every (∞,1)-category of (∞,1)-sheaves is characterized as being a sub-(∞,1)-topos Sh(S)PSh(S)Sh(S) \hookrightarrow PSh(S) of the (,1)(\infty,1)-topos of (∞,1)-presheaves on some (small) (∞,1)-category SS;

  • every such (,1)(\infty,1)-topos is a reflective (∞,1)-subcategory of PSh(S)PSh(S), hence a localization of an (∞,1)-category at a collection W={YX}W = \{Y \to X\} of morphisms which are sent to equivalences by the left adjoint of the inclusion;

  • and the sheaves in Sh(S)PSh(S)Sh(S) \hookrightarrow PSh(S) are precisely the local objects with respect to this collection WW of morphisms, i.e. precisely those objects APSh(S)A \in PSh(S) such that PSh(S)(X,A)PSh(S)(Y,A)PSh(S)(X,A) \to PSh(S)(Y,A) is an isomorphism in the homotopy category, which we shall write PSh(S)(X,A)PSh(S)(Y,A)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 YXY \to X in WW usually come from hypercovers YXY \to X;

in this case the above condition becomes PSh(S)(X,A)PSh(S)(colim ΔY ,A)PSh(S)(X,A) \stackrel{\simeq}{\to} PSh(S)(colim^\Delta Y_\bullet, A) which is equivalent to PSh(S)(X,A)lim ΔPSh(S)(Y ,A) PSh(S)(X,A) \stackrel{\simeq}{\to} lim^\Delta PSh(S)(Y_\bullet, A) . This in turn is usually equivalently written

A(X)Desc(Y X,A):=lim ΔA(Y ). A(X) \stackrel{\simeq}{\to} Desc(Y_\bullet \to X, A) := lim^\Delta A(Y_\bullet) \,.

And this is the form of the local object-condition which is usually called descent condition.

Descent for ordinary sheaves

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 SS any small category and Set the category of small sets, write PSh(S)=[S op,Set]\mathrm{PSh}(S) = [S^{op}, Set] for the category of presheaves on SS. Categories of this form enjoy various nice properties which are familiar from SetSet itself, and which are summarized by saying that PSh(S)\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 SS is a sub-topos of PSh(S)PSh(S) in that it is a full and faithful functor Sh(S)PSh(S)Sh(S)\hookrightarrow PSh(S) which is a geometric morphism.

One finds that the reflective subcategory Sh(S)PSh(S)Sh(S) \hookrightarrow PSh(S) of sheaves inside presheaves is the localization of PSh(S)PSh(S) at morphisms f:YXf : Y \to X called local isomorphisms, which are determined by and determing the choice of topos-inclusion. A presheaf AA is a sheaf precisely if it is a local object with respect to these local isomorphisms, that is precisely if

Hom PSh(S)(X,A)Hom PSh(S)(f,A)Hom PSh(S)(Y,A) Hom_{PSh(S)}(X,A) \stackrel{Hom_{PSh(S)}(f,A)}{\to} Hom_{PSh(S)}(Y,A)

is an isomorphism for all local isomorphisms ff.

This locality condition is in fact the descent condition: the sheaf has to descend from YY down to XX. More concretely, this condition is called a descent condition when evaluated on morphisms f:YXf : Y \to X which are hypercovers:

namely if π:Y 1X\pi : Y^1 \to X is a local epimorphism with respect to the coverage that corresponds to the localization and if π 2:Y 2Y 1× XY 1\pi_2 : Y^2 \to Y^1 \times_X Y^1 is a local epimorphism, then with

Y :=(Y 2Y 1) Y^\bullet := (Y^2 \stackrel{\to}{\to} Y^1)

being the two canonical morphisms out of Y 2Y^2, it follows that the canonical morphism

colimY X colim Y^\bullet \to X

is a local isomorphism.

(This is excercise 16.6 in Categories and Sheaves).

Therefore for a presheaf AA to be a sheaf, it is necessary that

Hom PSh(S)(X,A)Hom PSh(S)(colimY ,A) Hom_{PSh(S)}(X,A) \stackrel{\simeq}{\to} Hom_{PSh(S)}(colim Y^\bullet, A)

is an isomorphism. The colimit may be taken out of the hom-functor to make this equivalently

Hom PSh(S)(X,A)limHom PSh(S)(Y ,A). Hom_{PSh(S)}(X,A) \stackrel{\simeq}{\to} lim Hom_{PSh(S)}(Y^\bullet, A) \,.

It is convenient, suggestive and common to write A(X):=Hom PSh(S)(X,A)A(X) := Hom_{PSh(S)}(X,A), A(Y ):=Hom PSh(S)(Y ,A)A(Y^\bullet) := Hom_{PSh(S)}(Y^\bullet,A), following the spirit of the Yoneda lemma whether or not XX and/or Y Y^\bullet are representable. In that notation the above finally becomes

A(X)limA(Y ). A(X) \stackrel{\simeq}{\to} lim A(Y^\bullet) \,.

This is the form of the condition that is most commonly called the descent condition.

Descent for simplicial presheaves

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 SS, as follows.

Let [S op,SSet][S^{op}, SSet] be the SSet-enriched category of simplicial presheaves on SS.

Recall from model structure on simplicial presheaves that there is the global and the local injective simplicial model structure on [S op,SSet][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][S^{op}, SSet] with respect to the global injective model structure is (the SSet-enriched category realization of) the (,1)(\infty,1)-category PSh (,1)(S)PSh_{(\infty,1)}(S) of (∞,1)-presheaves on SS.

  • the full simplicial subcategory on fibrant-cofibrant objects of [S op,SSet][S^{op}, SSet] with respect to the local injective model structure is (the SSet-enriched category realization of) the (,1)(\infty,1)-category Sh¯ (,1)(S)\bar{Sh}_{(\infty,1)}(S) which is the hypercompletion of the (,1)(\infty,1)-category Sh (,1)(S)Sh_{(\infty,1)}(S) of (∞,1)-sheaves on SS.

Since with respect to the local or global injective model structure all objects are automatically cofibrant, this means that Sh¯ (,1)(S)\bar Sh_{(\infty,1)}(S) is the full sub-(,1)(\infty,1)-category of PSh (,1)(S)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:YXf : Y \to X in PSh (,1)(S)PSh_{(\infty,1)}(S) – hence between injective-fibrant objects in [S op,PSh(S)][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 YXY \to X a morphism of simplicial presheaves, the condition that a simplicial presheaf AA is local with respect to it, hence satisfies descent with respect to it, is generally that

RHom(X,A) RHom(Y,A) RHom(hocolim nY n,A) holim nRHom(Y n,A) =:holim nA(Y n) \begin{aligned} RHom(X,A) \stackrel{}{\to} & RHom(Y,A) \\ & \simeq RHom(hocolim_n Y_n, A) \\ & \simeq holim_n RHom(Y_n, A) \\ & =: holim_n A(Y_n) \end{aligned}

is a weak equivalence, where RHomRHom denotes the corresponding (,1)(\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 YY and AA are fibrant with respect to the given model structure.

The details on which morphisms YXY \to X one needs to check against here have been worked out in

  • D. Dugger, S. Hollander, D. Isaksen, Hypercovers and simplicial presheaves (pdf)

We now describe central results of that article.


For XSX \in S an object in the site regarded as a simplicial presheaf and Y[S op,SSet]Y \in [S^{op}, SSet] a simplicial presheaf on SS, a morphism YXY \to X is a hypercover if it is a local acyclic fibration, i.e. of for all VSV \in S and all diagrams

Λ k[n]V Y Δ nV XrespectivelyΔ nV Y Δ nV X \array{ \Lambda^k[n]\otimes V &\to & Y \\ \downarrow && \downarrow \\ \Delta^n\otimes V &\to& X } \;\; respectively \;\, \array{ \partial \Delta^n\otimes V &\to & Y \\ \downarrow && \downarrow \\ \Delta^n\otimes V &\to& X }

there exists a covering sieve {U iV}\{U_i \to V\} of VV with respect to the given Grothendieck topology on SS such that for every U iVU_i \to V in that sieve the pullback of the abve diagram to UU has a lift

Λ k[n]U i Y Δ nU i XrespectivelyΔ nU i Y Δ nU i X. \array{ \Lambda^k[n]\otimes U_i &\to & Y \\ \downarrow &\nearrow & \downarrow \\ \Delta^n\otimes U_i &\to& X } \;\; respectively \;\, \array{ \partial \Delta^n\otimes U_i &\to & Y \\ \downarrow &\nearrow& \downarrow \\ \Delta^n\otimes U_i &\to& X } \,.

If SS is a Verdier site then every such hypercover YXY \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 YXY \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 (,1)(S)Sh_{(\infty,1)}(S) – i.e. the fibrant objects with respect to the projective model structure on [S op,SSet][S^{op}, SSet] – are precisely those objects AA of PSh (,1)(S)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:YXf : Y \to X in [S op,SSet][S^{op}, SSet] we have that

[S op,SSet](X,A)[S op,SSet](Y,A) [S^{op}, SSet](X,A) \stackrel{\simeq}{\to} [S^{op}, SSet](Y,A)

is a weak equivalence of simplicial sets.


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 opSSetF : S^{op} \to SSet, which we may regard as a presheaf F:Δ op×S opSetF : \Delta^{op}\times S^{op} \to Set, is isomorphic to the weighted colimit

Fcolim ΔF F \simeq colim^\Delta F_\bullet

which is equivalently the coend

F [n]ΔΔ nF n, F \simeq \int^{[n] \in \Delta} \Delta^n \cdot F_n \,,

where F nF_n is the Set-valued presheaf of nn-cells of FF regarded as an SSetSSet-valued presheaf under the inclusion SetSSetSet \hookrightarrow SSet, and where the SSet-weight is the canonical cosimplicial simplicial set Δ\Delta, i.e. for all XSX \in S

F:X [n]ΔΔ n×F(X) n. F : X \mapsto \int^{[n] \in \Delta} \Delta^n \times F(X)_n \,.

In particular therefore for AA a Kan complex-valued presheaf the descent condition reads

[S op,SSet](X,A)[S op,SSet](colim ΔY ,A)lim Δ[S op,SSet](Y ,A). [S^{op}, SSet](X,A) \stackrel{\simeq}{\to} [S^{op}, SSet](colim^\Delta Y_\bullet,A) \simeq lim^\Delta [S^{op}, SSet](Y_\bullet,A) \,.

With the shorthand notation introduced above the descent condition finally reads, for all global-injective fibrant simplicial presheaves AA and hypercovers UXU \to X:

A(X)lim ΔA(Y ). A(X) \stackrel{\simeq}{\to} lim^\Delta A(Y_\bullet) \,.

The right hand here is often denoted Desc(Y X,A)Desc(Y_\bullet \to X, A), in which case this reads

A(X)Desc(Y X,A). A(X) \stackrel{\simeq}{\to} Desc(Y_\bullet \to X, A) \,.

Descent for strict ω\omega-groupoid valued presheaves

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

CrsCmplxStrωGrpdNSSet. CrsCmplx \stackrel{\simeq}{\to} Str \omega Grpd \stackrel{N}{\to} SSet \,.

Since for instance something as simple as an abelian group AA regarded as a complex of groups in degree nn (hence as an nn-group) already bcomes a somewhat involved object to understand under the nervet operation,

it is desireable to have a means to control descent for simplicial presheaves which happen to factor through the ω\omega-nerve directly in the context of StrωCatStr \omega Cat.

In his work on descent

Ross Street considered presheaves with values in strict ω-categories

A:S opStrωCat A : S^{op} \to Str \omega Cat

and declared the descent ω\omega-category with respect to a simplicial object Y :Δ opSY_\bullet : \Delta^{op} \to S to be the weighted limit in StrωCatStr\omega-Cat-enriched category theory

lim FΔA(U ), lim^{F \Delta} A(U^\bullet) \,,

where O:=FΔ:ΔStrωCatO := F \Delta : \Delta \to Str \omega Cat are the orientals, i.e. the free ω\omega-categories on the simplicial simplices

O=FΔ, O = F \circ \Delta \,,

where F:SSetStrωCatF : SSet \to Str\omega Cat is the right adjoint to the ω-nerve N:StrωCatSSetN : Str \omega Cat \to SSet.

The two precscriptions

lim ΔNA(U ) inSSet lim FΔA(U ) inStrωGrpd \array{ lim^\Delta N A(U^\bullet) & in SSet \\ \\ lim^{F \Delta} A(U^\bullet) & in Str \omega Grpd }

have a very similar appearance. The following theorem asserts if and when they are actually equivalent.

Theorem (Dominic Verity)

There exists a canonical comparison map

N(Desc Street(U ,A)):=N(lim FΔA(U ))Desc simp(U ,NA):=holim N(A(U )). N(Desc_{Street}(U^\bullet, A)) := N(lim^{F \Delta} A(U^\bullet)) \;\;\;\stackrel{\;\;\;\;\;\;\;\;\;}{\hookrightarrow}\;\;\; Desc_{simp}(U^\bullet, N \circ A) := holim_\bullet N(A(U^\bullet)) \,.

This is a weak equivalence of Kan complexes if the cosimplicial simplicial set N(A(U ))N(A(U^\bullet)) is Reedy fibrant.

Descent in terms of gluing conditions

We unwrap the expression

lim ΔA(Y ) lim^\Delta A(Y_\bullet)

for the descent data for a presheaf AA with respect to a (hyper)cover YXY \to X

This weighted limit (whether taken in SSetSSet- or in StrωCatStr \omega Cat-enriched category theory) is given by the coend

lim WΔA(Y ) [n]Δ[Δ n,A(Y n)]. lim^W\Delta A(Y_\bullet) \simeq \int^{[n] \in \Delta} [\Delta^n, A(Y_n)] \,.

Unwrapping what this means one finds that an object/vertex of this is a choice of nn-simplex in each A(Y n)A(Y_n), subject to conditions which say that the boundary of this nn-simplex must be obtained from pullback of AA along the maps Y nY [n1]Y_n \to Y_{[n-1]} of the (n1)(n-1)-simplex in A(Y n1)A(Y_{n-1})

Namely an object in

lim WΔA(Y ) [n]Δ[Δ n,A(Y n)] lim^W\Delta A(Y_\bullet) \simeq \int^{[n] \in \Delta} [\Delta^n, A(Y_n)]

is a commuting diagram

[2] Δ 2 f A(Y 2) [1] Δ 1 g A(Y 1) [0] Δ 0 a A(Y 0) \array{ \uparrow &&&& \uparrow && \uparrow \\ [2] &&&& \Delta^2 &\stackrel{f}{\to}& A(Y_2) \\ \uparrow &&&& \uparrow && \uparrow \\ [1] &&&& \Delta^1 &\stackrel{g}{\to}& A(Y_1) \\ \uparrow &&&& \uparrow && \uparrow \\ [0] &&&& \Delta^0 &\stackrel{a}{\to}& A(Y_0) }

where the vertical arrows indicate all the simplicial maps of the cosimplicial objects Δ\Delta and A(Y )A(Y_\bullet).

So this is

  • on Y 0Y_0 an object aA(Y 0)a \in A(Y_0);

  • on “double intersections” (might be a cover of double intersections) Y 1Y_1 a gluing isomorphism g:π 1 *aπ 2 *ag : \pi_1^* a \to \pi_2^* a which identifies the two copies of aa obtained by pullback along the two projection maps π 1,π 2:U× XUU\pi_1, \pi_2 : U \times_X U \to U.

  • on “triple intersections” Y 2Y_2 a gluing 2-isomorphism π 2 *a π 12 *g f π 23 *g π 1 *a π 13 *g π 3 *a \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.

Gluing for ordinary stacks

The article

  • Sharon Hollander, A Homotopy Theory for Stacks (arXiv)

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

[B(),A]:S opSSet, [B(-), A] : S^{op} \to SSet \,,

where B:S[S op,SSet]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

B:=Π:U(VS(V×Δ ,U)) B := \Pi : U \mapsto (V \mapsto S(V \times \Delta^\bullet, U))

in which case the presheaf

[Π(),A] [\Pi(-),A]

would assign to USU \in S the pre-\infty-stack of “trivial AA-principial bundles with flat connection”.

For YXY \to X a given (hyper)cover, the descent object for [B(),A][B(-), A] can be expressed as

Desc(Y X,[B(),A]) :=lim Δ[B(Y ),A] [n]Δ[Δ n,[B(Y n),A]] [n]Δ[Δ nB(Y n),A] [ [n]ΔΔ nB(Y n),A] [colim ΔB(Y ),A]. \begin{aligned} Desc(Y_\bullet \to X, [B(-), A]) & := lim^\Delta [B(Y_\bullet),A] \\ & \simeq \int^{[n]\in \Delta} [ \Delta^n, [B(Y_n),A] ] \\ & \simeq \int^{[n]\in \Delta} [ \Delta^n \otimes B(Y_n), A] \\ & \simeq [\int_{[n] \in \Delta} \Delta^n \otimes B(Y_n)\;,\; A ] \\ & \simeq [colim^\Delta B(Y_\bullet), A] \end{aligned} \,.

This way the descent for [B(),A][B(-),A] on the object U=colim ΔU U = colim^\Delta U_\bullet is reexpressed as descent for AA of the BB-modified object colim ΔB(Y )colim^\Delta B(Y_\bullet). Following Street, this we may call the codescent object, as it co-represents descent.

Monadic 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.


A connection between the monadic descent and the descent in the language of fibered categories is proved in

  • Jean Bénabou, Jacques Roubaud, Monades et descente, C. R. Acad. Sc. Paris, t. 270 (12 Janvier 1970), Serie A, 96–98, (link, Bibliothèque nationale de France)

Revised on May 22, 2015 07:23:44 by Zoran Škoda (