# Contents

## 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 identifiy 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.

See descent.

### For strict $\omega$-category-valued presheaves

In (Street) a proposal for a definition of descent objects for presehaves with values in strict ω-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 $\Desc [N(A),X]\cong[A,X]$

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