What is descent data (of higher categories), conceptually?

Question

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a quasi-coherent sheaf on each $U_i$ and $\phi_{ij}$ is an isomorphism $pr_2^*\mathcal{E}_j\to pr_1^*\mathcal{E}_i$ in $Qcoh(U_{ij})$ which satisfies the cocycle condition $$ pr_{13}^*\phi_{ik}=pr_{12}^*\phi_{ij}\circ pr_{23}^*\phi_{jk}. $$ We can make the descent data into a category where the morphisms are those compatible with the $\phi_{ij}$'s. Moreover we have an equivalence of categories $$ \text{Desc}(X,\mathcal{U})\simeq Qcoh(X). $$

Of course the definition of descent data has various generalizations. For example instead of the category of quasi-coherent sheaves we can consider an arbitrary fibered category $\mathcal{F}$ over spaces. For any map between spaces $\pi: U\to X$ we have a cosimplicial diagram of categories $$ \mathcal{F}(X)\to \mathcal{F}(U)\rightrightarrows \mathcal{F}(U\times_X U)\ldots $$ An object with descent data on is a collection $(\mathcal{E},\phi)$ where $\mathcal{E}$ is an object in $\mathcal{F}(U)$ and $\phi$ is an isomorphism $pr_2^*\mathcal{E}\to pr_1^*\mathcal{E}$ which satisfies the cocycle condition $$ pr_{13}^*\phi=pr_{12}^*\phi\circ pr_{23}^*\phi. $$ We notice that the category $\text{Desc}(U\to X)$ is equivalent to the fiber product of categories $\mathcal{F}(U)\times_{\mathcal{F}(U\times_X U)} \mathcal{F}(U)$ (as Zhen Lin points out in the comment, we need to also consider $\mathcal{F}(U\times_X U\times_X U)$).

We also notice that if $\pi$ is flat then $\phi$ can be also expressed as the comodule structure $\mathcal{E}\to \pi^*\pi_*\mathcal{E}$ since $\pi^*\pi_*\mathcal{E}\cong (pr_2)_*pr_1^*\mathcal{E}$. Moreover in the flat case (together with some condition on $F$ I guess) we have $$ \text{Desc}(U\to X)\simeq \mathcal{F}(X). $$

Now we consider the case that $\mathcal{F}$ contains some higher structure. In this case we have an augmented cosimplicial diagram of (higher) categories $$ \mathcal{F}(X)\to \mathcal{F}(U)\rightrightarrows \mathcal{F}(U\times_X U)\ldots $$ and the definition of descent data should be modified. For example in the recent version of Yekutieli's Twisted Deformation Quantization of Algebraic Varieties Section 5, an explicit definition of descent data of cosimplicial cross groupoid is given.

Before given the explicit construction, I would like to know what SHOULD the descent data be. Or more precisely what are the properties that descent data must satisfy?

One attempt is to say that descent data are extra structures (say comodule) on $\mathcal{F}(U)$ such that we have the (weak) equivalence $\text{Desc}(U\to X)\simeq \mathcal{F}(X)$, but the equivalence only exists in good (say flat) cases while the descent data can be given in general cases.

Therefore is the alternative description better? The category of descent data should be the homotopy limit (or totalization) of the cosimplicial diagram $$ \mathcal{F}(U)\rightrightarrows \mathcal{F}(U\times_X U)\ldots $$

Answer 1

The category of descent data is indeed the homotopy limit of your cosimplicial diagram. In the case where $\mathcal{F}$ actually is fibred in categories (and not higher categories), then you can truncate above degree 2, recovering the classical definition. If $\mathcal{F}$ is fibred in sets, then you can even truncate above degree 1, recovering the classical sheaf condition. So, morally, the category of descent data generalises the set of matching elements of a presheaf with respect to a cover.

As for where the definition comes from: well, it comes from the observation that a morphism $p : X \to Y$ is an effective epimorphism in the $(\infty, 1)$-category of $\infty$-groupoids if and only if it fits into a colimit cocone for the Čech complex of $p : X \to Y$ (thought of as a diagram of shape $\mathbf{\Delta}^\mathrm{op}$). (This is more or less by definition.) Thus, $p : X \to Y$ is an effective epimorphism if and only if the induced morphism $p^* : [Y, Z] \to [X, Z]$ of mapping spaces fits into a limit cone for the obvious diagram of shape $\mathbf{\Delta}$ for every $\infty$-groupoid $Z$, i.e. if and only if $[-, Z]$ satisfies the descent condition with respect to $p : X \to Y$.

Answer 2

Before given the explicit construction, I would like to know what SHOULD the descent data be. Or more precisely what are the properties that descent data must satisfy?

Zhen Lin's answer is, of course, correct, but it's a bit terse, so here is some elaboration.

Suppose $f : X \to Y$ is a map of sets and that you want to know a real-valued function $r : Y \to \mathbb{R}$. However, you aren't given $r$, but only its pullback $r \circ f : X \to \mathbb{R}$ along $f$. When can you always recover $r$ from this information, and how do you identify when such a function $s : X \to \mathbb{R}$ is a pullback along $f$? The answers, of course, are

• Iff $f$ is surjective, and
• Iff $s$ respects the equivalence relation on $X$ determined by $f$ (where $x_1 \sim x_2$ iff $f(x_1) = f(x_2)$).

This situation can be cast in more categorical language as follows. In a category with pullbacks, given a morphism $f : X \to Y$ we can take its kernel pair, which is the pullback $X \times_Y X$. In $\text{Set}$, this is

$$X \times_Y X = \{ (x_1, x_2) \in X^2 : f(x_1) = f(x_2) \}$$

and hence it, together with the two projection maps to $X$, exactly reproduces the equivalence relation on $X$ determined by $f$ alluded to above. The kernel pair is so named because it generalizes the kernel to nonabelian situations. In any case, a natural thing to try to do with the two projections $X \times_Y X \rightrightarrows X$ is to take their coequalizer. If $f : X \to Y$ is already this coequalizer, $f$ is said to be an effective epimorphism. Our discussion for $\text{Set}$ implicitly uses the fact that every epimorphism of sets is effective.

The significance of effective epimorphisms is that the following version of descent holds for them: suppose $F$ is a contravariant functor from the ambient category to some other category which sends coequalizers to equalizers (in particular, any representable presheaf). Then $F(Y)$ is the equalizer of the pair of maps

$$F(X) \rightrightarrows F(X \times_Y X)$$

given by applying $F$ to the kernel pair projections. Our discussion for $\text{Set}$ says precisely this for the special case $F(-) = \text{Hom}(-, \mathbb{R})$.

Descent is the higher version of this story where the kernel pair is replaced by the Cech complex and $F$ takes values in a higher category. The reason it is not just a formal exercise involving functors which send homotopy colimits to homotopy limits is that one needs to identify, in a given higher category, which morphisms are effective in the relevant sense.