nLab descent object

Idea

Quite generally, one says that an object $A$ in a category or higher category $\mathcal{C}$ satisfies descent along a given morphism $p : \hat X \to X$ in $\mathcal{C}$ if it is a $p$-local object, hence if the induced map – the descent morphism

$$\mathcal{C}(X, A) \to \mathcal{C}(\hat X, A)$$

is an equivalence. We may read this as saying that every collection of $A$-data on $\hat X$ “descends” down along $p$ to $X$.

In the context the hom object $\mathcal{C}(\hat X, A)$ is also called the descent object.

While roughly synonyms, typically one speaks of “descent” instead of locality when $\mathcal{C}$ is a category of presheaves or higher presheaves ((2,1)-presheaves, (∞,1)-presheaves, (∞,n)-presheaves).

In this case, in turn, the objects $X$ above are typically representables of a given site (or higher site) and $\hat X$ is either the Cech nerve of a covering family with respect to a chosen coverage/Grothendieck topology, or is the colimit of this Čech nerve: the corresponding sieve (the codescent object).

The descent condition then says that the presheaf $X$ satisfies the sheaf-condition (stack-condition, (∞,1)-sheaf/∞-stack-condition, etc.) for this given covering family.

Whether one takes $\hat X$ to be the Cech nerve or the corresponding sieve depends on homotopical details of the setup. If $\mathcal{C}$ is taken to be an (∞,1)-category, then it typically does not matter. But if $\mathcal{C}$ is instead just a homotopical category presenting the desired higher category, then $\hat X$ needs to satisfy some extra conditions (such as cofibrancy) to ensure that $\mathcal{C}(\hat X, A)$ is indeed the correct descent object, and not too small.

For instance when working with the injective model structure on simplicial presheaves, every object is cofibrant and we can take $\hat X$ to be the sieve. But when working with the projective model structure then (as discussed there) $\hat X$ needs to be split, which means that we need to use the Cech nerve and even ensure that the corresponding covering family behaves like a good cover (or, more generally, form a split hypercover).

Details

For ordinary presheaves

For ordinary presheaves, a descent object is a set of matching families

More in detail, let $C$ be a site, let $X \in C$ be an object, $\{U_i \to X\}$ a covering family and $S(\{U_i\}) \hookrightarrow X$ the corresponding sieve.

Then for $A : C^{op} \to Set$ any presheaf on $C$, the descent object with respect to this covering is the hom set

$$Desc(\{U_i\}, A) = PSh(S(\{U_i\}), A) \,.$$

This is discussed in detail at sheaf, so just briefly:

the sieve may be realized as the coequalizer

$$\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i \to S(\{U_i\}) \,.$$

Accordingly the hom out of this realizes the descent object as the equalizer

$$Desc(\{U_i\}, A) \to \prod_{i} A(U_i) \stackrel{\to}{\to} \prod_{i, j} A(U_i \cap U_j) \,.$$

Writing this out in components shows that this is the set of matching families.

If the descent morphism

$$[C^{op}, Set](X, A) \to Desc(\{U_i\}, A)$$

is an isomorphism one says that $A$ satisfies the sheaf-condition with respect to the cover $\{U_i \to X\}$. If this morphism is only a monomorphism one says that $A$ satisfies the separated presheaf-condition.

For groupoid valued presheaves / pseudofunctors

For $A : C^{op} \to$ Grpd a 2-functor (hence a “pseudofunctor” if $C$ is an ordinary category regarded as a 2-category) and for $\hat X \to X$ a covering morphism in $C$, the descent object now is a groupoid

$$Desc(\hat X, A) := [C^{op}, Grpd](\hat X, A) \in Grpd \,.$$

If the descent morphism

$$[C^{op}, Grpd](X, A) \to Desc(\hat X, A)$$

is an equivalence of groupoids, one says that $A$ satisfies the (2,1)-sheaf- or stack-condition with respect to the cover $\hat X \to X$. If it is just a full and faithful functor, one says (sometimes) that $A$ satisfies the condition for a separated prestack with respect to this cover.

Similar statements hold for the case of 2-functors with values in Cat. Here one also often talks about a stack-condition, though less ambiguous would be to speak of 2-sheaf-conditions.

By the Grothendieck construction one may identify pseudofunctors $C^{op} \to Cat$ equivalently with fibered categories (or just categories fibered in groupoids for $C^{op} \to Grpd$) over $C$, and all of the above has analogs in this dual description.

For simplicial presheaves

See descent.

For strict $\omega$-category-valued presheaves

In (Street) a proposal for a definition of descent objects for presehaves with values in strict ∞-categories was proposed. Additional homotopical conditions to ensure that this gives the right answer were discussed in (Verity).

Definition

Let $C$ be a category, let $E_1\stackrel{\stackrel{d_1}{\to}}{\stackrel{d_0}{\to}}E_0\xrightarrow{p}B$ be morphisms where the parallel arrows $\mathcal{E}:=\{d_0,d_1:E_1\to E_0\}$ are seen as a diagram, let $X\in C_0$ be an object.

Applying the functor $C(-,X)$ to this sequence gives

$$C(E_1,X)\xleftarrow{C(d_0,X),C(d_1,X)}C(E_0,X)\xleftarrow{C(p,X)}C(B,X)$$

If this diagram is for all $X\in C_0$ an equalizer diagram $B$ is called codescent object for the diagram $\mathcal{E}$.

Definition

Let $E_0\to\E_1\to E_2$ be a diagram where $E_0\xrightarrow{\partial_0}E_1\xrightarrow{\partial_0}E_2$, $E_0\xleftarrow{\iota_0}E_1\xrightarrow{\partial_1}E_2$, $E_0\xrightarrow{\partial_1}E_1\xrightarrow{\partial_2}E_2$ satisfying $\partial_s\partial_r=\partial_r\partial_{s-1}$ for $r\lt s$ and $\iota_0\partial_0=\iota_0\partial_1$ (these are the identities characterizing a truncated cosimplicial category).

Then the descent category $\Desc E$ of $E$ has as objects pairs $(F,f)$ where $F\in E_0$, $f:\partial_1 F\to \partial_0 F$ such that $\iota_0 f=\id_F$ and $\partial_0 f=\partial_2( f)\circ \partial_0 (f)$ and a morphism $(F,f)\to (G,g)$ consists of a morphism $(u:F\to G)\in E_1$ such that $\partial_0 u\circ f=g\circ \partial_1 u$.

Instance

Let $A$, $X$ be categories.

Then $\Desc [N(A),X]\cong[A,X]$

Related concepts

See also descent and category of descent data.

Discussion related to the computation of descent objects is also at model structure on cosimplicial simplicial sets.

References

See also the references at descent.

A definition of descent objects for presheaves with values in strict $\omega$-categories was proposed in

• Ross Street, Categorical and combinatorial aspects of descent theory

A discussion of a missing condition on this definition is in

• Dominic Verity, Relating descent notions (pdf nLab)