# Contents

## Idea

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

$𝒞\left(X,A\right)\to 𝒞\left(\stackrel{^}{X},A\right)$\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 $\stackrel{^}{X}$descends” down along $p$ to $X$.

In the context the hom object $𝒞\left(\stackrel{^}{X},A\right)$ is also called the descent object.

While roughly synonyms, typically one speaks of “descent” instead of locality when $𝒞$ 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 $\stackrel{^}{X}$ 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 $\stackrel{^}{X}$ to be the Cech nerve or the corresponding sieve depends on homotopical details of the setup. If $𝒞$ is taken to be an (∞,1)-category, then it typically does not matter. But if $𝒞$ is instead just a homotopical category presenting the desired higher category, then $\stackrel{^}{X}$ needs to satisfy some extra conditions (such as cofibrancy) to ensure that $𝒞\left(\stackrel{^}{X},A\right)$ 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 $\stackrel{^}{X}$ to be the sieve. But when working with the projective model structure then (as discussed there) $\stackrel{^}{X}$ 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, $\left\{{U}_{i}\to X\right\}$ a covering family and $S\left(\left\{{U}_{i}\right\}\right)↪X$ the corresponding sieve.

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

$\mathrm{Desc}\left(\left\{{U}_{i}\right\},A\right)=\mathrm{PSh}\left(S\left(\left\{{U}_{i}\right\}\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$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

$\coprod _{i,j}{U}_{i}\cap {U}_{j}\stackrel{\to }{\to }\coprod _{i}{U}_{i}\to S\left(\left\{{U}_{i}\right\}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\mathrm{Desc}\left(\left\{{U}_{i}\right\},A\right)\to \prod _{i}A\left({U}_{i}\right)\stackrel{\to }{\to }\prod _{i,j}A\left({U}_{i}\cap {U}_{j}\right)\phantom{\rule{thinmathspace}{0ex}}.$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

$\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\left(X,A\right)\to \mathrm{Desc}\left(\left\{{U}_{i}\right\},A\right)$[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 $\left\{{U}_{i}\to X\right\}$. If this morphism is only a monomorphism one says that $A$ satisfies the separated presheaf-condition.

### For groupoid valued presheaves / pseudofunctors

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

$\mathrm{Desc}\left(\stackrel{^}{X},A\right):=\left[{C}^{\mathrm{op}},\mathrm{Grpd}\right]\left(\stackrel{^}{X},A\right)\in \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$Desc(\hat X, A) := [C^{op}, Grpd](\hat X, A) \in Grpd \,.

If the descent morphism

$\left[{C}^{\mathrm{op}},\mathrm{Grpd}\right]\left(X,A\right)\to \mathrm{Desc}\left(\stackrel{^}{X},A\right)$[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 $\stackrel{^}{X}\to X$. 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 ${C}^{\mathrm{op}}\to \mathrm{Cat}$ equivalently with fibered categories (or just categories fibered in groupoids for ${C}^{\mathrm{op}}\to \mathrm{Grpd}$) 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 ${E}_{1}\stackrel{\stackrel{{d}_{1}}{\to }}{\stackrel{{d}_{0}}{\to }}{E}_{0}\stackrel{p}{\to }B$ be morphisms where the parallel arrows $ℰ:=\left\{{d}_{0},{d}_{1}:{E}_{1}\to {E}_{0}\right\}$ are seen as a diagram, let $X\in {C}_{0}$ be an object.

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

$C\left({E}_{1},X\right)\stackrel{C\left({d}_{0},X\right),C\left({d}_{1},X\right)}{←}C\left({E}_{0},X\right)\stackrel{C\left(p,X\right)}{←}C\left(B,X\right)$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 $ℰ$.

###### Definition

Let ${E}_{0}\to {E}_{1}\to {E}_{2}$ be a diagram where ${E}_{0}\stackrel{{\partial }_{0}}{\to }{E}_{1}\stackrel{{\partial }_{0}}{\to }{E}_{2}$, ${E}_{0}\stackrel{{\iota }_{0}}{←}{E}_{1}\stackrel{{\partial }_{1}}{\to }{E}_{2}$, ${E}_{0}\stackrel{{\partial }_{1}}{\to }{E}_{1}\stackrel{{\partial }_{2}}{\to }{E}_{2}$ satisfying ${\partial }_{s}{\partial }_{r}={\partial }_{r}{\partial }_{s-1}$ for $r and ${\iota }_{0}{\partial }_{0}={\iota }_{0}{\partial }_{1}$ (these are the identities characterizing a truncated cosimplicial category).

Then the descent category $DescE$ of $E$ has as objects pairs $\left(F,f\right)$ where $F\in {E}_{0}$, $f:{\partial }_{1}F\to {\partial }_{0}F$ such that ${\iota }_{0}f={id}_{F}$ and ${\partial }_{0}f={\partial }_{2}\left(f\right)\circ {\partial }_{0}\left(f\right)$ and a morphism $\left(F,f\right)\to \left(G,g\right)$ consists of a morphism $\left(u:F\to G\right)\in {E}_{1}$ such that ${\partial }_{0}u\circ f=g\circ {\partial }_{1}u$.

###### Instance

Let $A$, $X$ be categories.

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

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