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Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried navigating the vast sea of literature).

I'm trying to start learning descent theory, and after seeing how descent along open covers for topological bundles already loosens the sheaf condition and adds to cocycle condition, I'm looking for a conceptual definition with machinery that encodes all the coherence conditions.

This MO question and its answers affirm that descent data is the homotopy limit of the the cosimplicial diagram obtained as the image of the Čech nerve along $\mathcal F$. However, this already leaves me unsure as to what $\mathcal F$ should be in the general setting. For sheaves, it's just a presheaf of sets. For bundles, it seems to be a category fibered in categories (thought of as a presheaf of categories?).

Below are two similar remarks which involve something I don't understand. First, from Zhen Lin's answer to the question linked above:

The category of descent data is indeed the homotopy limit of your cosimplicial diagram. In the case where $F$ actually is fibred in categories (and not higher categories), then you can truncate above degree 2, recovering the classical definition. If $F$ 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.

Second, by David Roberts, in the first comment to this MO question:

If you're interested in $n$-limits, these are more easily treated using homotopy colimits, as we don't understand the $n$-category of $(n−1)$-categories very well, for $n>3$, say. A coequaliser is the homotopy colimit (in a 1-category!) of a simplicial diagram truncated all the way down to a pair of parallel arrows. The "2-" version of this is a slightly less truncated simplicial diagram. Mapping out of these (in an appropriate model structure, in the absence of a good higher category structure) gives the cosimplicial version Street has, and replaces colimits by limits.

The three (main) things I don't understand are:

  1. Why can we "truncate"? The remarks suggest that if $\mathcal F$ is a presheaf taking values in $n$-categories (with $n=0$ being sets), then we "can truncate" the cosimplicial diagram above level $n+1$. I take it this means the (proper notion of a) limit of the entire cosimplicial diagram is the same as that of the truncated one, but I don't understand why this is true.
  2. Why is 'homotopy limit' the correct term here? I have some vague understanding that homotopy limits in the sense of derived functors of functors between homotopical categories somehow present $(\infty,1)$-limits, but this after choosing weak equivalences! Nowhere in the first question are weak equivalences mentioned, so what are they?
  3. The meaning of homotopy limit seems to change depending on context in David Robert's remark: coequalizers, which are just $1$-limits, are said to be homotopy limits of parallel pairs, which suggests the weak equivalences here on $\mathsf{Cat}$ are just taken to be isomorphisms. However, if we're fibered over categories then this no longer makes any sense...
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    $\begingroup$ For (1) I think the essential point is that the inclusion of $\Delta_{\le n}$ into $\Delta$ is "n-(co?)final" in the sense that the $\Delta_{\le n} \times_{\Delta} \Delta_{/[m]}$ has an $n$-connected nerve for all $m$. So the analog of Quillen's theorem A says that computing limits over one or the other should be the same. $\endgroup$ – Dylan Wilson Apr 18 '16 at 20:45
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    $\begingroup$ For (2) the weak equivalences are, of course, equivalences of $n$-categories (or $n$-groupoids). $\endgroup$ – Denis Nardin Apr 18 '16 at 21:01
  • $\begingroup$ @DylanWilson I'm only familiar with the most basic simplicial notions, so a more detailed answer or a reference would help me very much. Also, if I understand correctly, this blog post says the map you mention is not cofinal. $\endgroup$ – Arrow Apr 18 '16 at 21:29
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    $\begingroup$ @Arrow I think Dylan means "n-cofinal", that is cofinal in the world of n-categories. For example if you take the limit of a cosimplicial diagram in a 1-category (for example sets) this is the same thing as taking the equalizer of the first two arrows. Anyway I take your point, if I have time this evening I'll write a detailed answer if no one beats me to it :). $\endgroup$ – Denis Nardin Apr 18 '16 at 21:43
  • $\begingroup$ @DylanWilson does this perhaps follow easily from existence of a coskeleton functor? Right adjoints are always final... $\endgroup$ – Arrow Apr 30 '16 at 14:39
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To answer your question I'll need to do a fairly long digression on homotopy limits and colimits. Before I delve deep into the topic let me say that there's more than one way to describe this topic, for example some people like model categories, other people might prefer triangulated categories (shudder), I'll simply explain the point of view that has been most helpful for me in the past.

The language of $\infty$-categories

I think that one of the best way to understand homotopy limits and colimits is in the settings of $\infty$-categories. In fact, in the theory $\infty$-categories then homotopy limits and colimits are just like ordinary limits and colimits and I will stop prefixing everything with the word "homotopy" from now on.

My favourite catchphrase is that $\infty$-categories are just categories together with a notion of homotopy beteween morphisms. In fact you need a bit more, basically for every two objects $X$ and $Y$ and every $n\ge0$ you need a set $\textrm{Map}(X,Y)_n$ that morally corresponds to maps $X\times \Delta^n\to Y$ (the so-called higher homotopies) satisfying a set of compatibilities that I will not detail here (a good choice is to ask that $\textrm{Map}(X,Y)_\bullet$ form a Kan complex). To keep this concrete, examples are

  • Spaces, with the obvious definition of homotopy;
  • Manifolds and embeddings, with the notion of homotopy given by isotopies;
  • Chain complexes with the notion of chain homotopy;
  • Categories, where an homotopy between two functors is a natural isomorphism (and higher homotopies just chains of $n$ composable natural isomorphisms $F_0\to F_1\to\cdots\to F_n$).

(The last example is of course the most relevant for your question).

The important part about $\infty$-categories is that the notion of homotopy should seep through in our definition of commutative diagrams (so-called coherently commutative diagrams or coherent diagrams for short). For example a commutative square

$$\require{AMScd} \begin{CD} A @>{f}>> B\\ @V{h}VV @VV{g}V\\ C @>>{k}> D \end{CD}$$

is the datum of four objects, four maps and a homotopy between $gf$ and $kh$.

Once you have set up all of this you can go on and define initial and terminal objects, limits, colimits and so on and so forth exactly like you did for ordinary categories (that is without homotopies). These are often called "homotopy limits" and "homotopy colimits" to distinguish from the limits and colimits computed without paying attention to the homotopies.

Ok, but what's the deal with the Čech complex anyway?

Let's give an example. Suppose that $C,D$ are two categories and $F,G:C\to D$ two functors. If we ignored the homotopies the equalizer would simply be the objects $c\in C$ such that $Fc=Gc$. But we all know that this is a stupid notion. Naturally equivalent functors will give different answers and we do not want to distinguish between naturally equivalent functors. So we use the notion of equalizer in the $\infty$-category of categories.

As you can see immediately from studying the coherent diagrams of the form $*\to C\rightrightarrows D$ the objects of the equalizer in this brave new setting are objects $c\in C$ together with an isomorphism $\alpha:Fc\cong Gc$. Now that's a much better behaved notion!

Similarly, you can see that if $F:\Delta^{op}\to C$ is a functor to an ordinary category we get that

$\lim_{\Delta^{op}} F = \textrm{eq}(F([0])\rightrightarrows F([1]))$

so that $F(X)\to \lim F(\check U_\bullet)$ being an equivalence is just the ordinary sheaf condition. But what happens if we go one categorical level up? Now let $F:\Delta^{op}\to \mathrm{Cat}$ be a functor to categories. Then you can see that an object of $\lim_{\Delta^{op}}F$ is an object $x\in F([0])$ together with an isomorphism $\alpha:d_0^*x\cong d_1^*x$ in $F([1])$ such that $d_0^*\alpha\circ d_2^*\alpha = d_1^*\alpha$. But this is exactly the notion of descent datum.

So the notion of sheaf in this brave new context is just the clasical notion of stack

You might have noticed that we are using just a small portion of the diagram. This is because our categories do not have many "interesting" homotopies. Sets have no homotopies at all and categories have only interesting 1-homotopies, higher homotopies are basically composable sequences of 1-homotopies. In general, the higher the order of the interesting homotopies, the more pieces of $\Delta$ we have to use. (precisely for an $n$-category you just need to go to $[n+1]$. The reason for this is basically Quillen's theorem A, as noted by Dylan Wilson in the comments). For spaces and chain complexes you need to consider the whole of $\Delta$.

Conclusions and references

Ugh that's a long answer. I hope I managed to give at lesat an inking of what's going on without getting bogged by the technical details. If technical details are what you want however the standard references for $\infty$-categories are Luries's Higher Topos Theory and Higher Algebra. They are not an easy read, by anyone's standards.

The model of $\infty$-categories I've used is called fibrant simplicial categories. I think it is an excellent model to develop some intuition, but it is actually quite bad to work with. Most of the people in the industry use quasicategories, which are quite pleasant to work with but I didn't have the time to properly introduce them here.

Appendix: why can we truncate?

Let $E$ be an $n$-category, that is an $\infty$-category such that the mapping spaces $\mathrm{Map}(x,y)$ are $n$-truncated for every $x,y\in E$. Let $j:C\to D$ be a functor such that for every $d\in D$ the geometric realization $|C\times_D D_{d/}|$ is $n$-connected. Then for every functor $F:D\to E$ $\lim_D F$ exists if and only if $\lim_C Fj$ exists and they coincide.

Lemma 1: Let $K$ be a simplicial set such that the geometric realization $|K|$ is $n$-connected. Then for every $e\in E$ the limit of the constant functor $K\to E$ at $e$ exists and coincides with $e$.

Proof: $\mathrm{Map}(e',\lim_K e) \cong \lim_K \mathrm{Map}(e',e) = \mathrm{Map}(K,\mathrm{Map}(e',e))=\mathrm{Map}(e',e)$.

Lemma 2: Let $p:\tilde D\to D$ be a cartesian fibration such that for every $d\in D$ the geometric realization $|\tilde D_d|$ is $n$-connected. Then for every functor $F:D\to E$ the limit $\lim_D F$ exists if and only if the limit $\lim_{\tilde D} Fp$ exists and they coincide.

Proof: By an easy cofinality argument the right Kan extension along a cartesian fibration is obtained by computing the limits fiberwise. Then the thesis follows from Lemma 1.

Proof of the main result: Let $\tilde D\to D$ the cartesian fibration classified by the functor $D^{op}\to \mathrm{Cat}$ given by $d\mapsto C\times_D D_{d/}$. We have a canonical functor $C\to \tilde D$ sending $c$ to $(c,jc=jc)$. A standard cofinality argument implies that the functor $C\to \tilde D$ is coinitial. Then the thesis follows from the previous lemma.

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  • $\begingroup$ If someone can fix the commutative diagram I will be grateful... $\endgroup$ – Denis Nardin Apr 19 '16 at 0:24
  • $\begingroup$ Thank you very much for the answer. I looked in 'Higher Topos Theory' but could not find any statement of the $n$-cofinality of the inclusion Dylan Wilson mentioned (in fact I could not find a definition of $n$-cofinality, only of final maps of simplicial sets). Could you please help me with the details? $\endgroup$ – Arrow Apr 19 '16 at 9:04
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    $\begingroup$ There is no such in HTT. I remember trying to find a reference for this and failing a while ago, but have dim memories of Lurie getting around this problem a different way... I'll see if I can find it. $\endgroup$ – Dylan Wilson Apr 19 '16 at 10:59
  • $\begingroup$ I think I should be able to cobble together a proof today. If anyone finds a genuine reference it will be better of course. Unfortunately I've looked in HTT, HA and Joyal's notes on quasicategories without success. (and thanks @ZhenLin for the diagram :)) $\endgroup$ – Denis Nardin Apr 19 '16 at 11:21
  • $\begingroup$ Thank you! I might ask this question separately in a few hours. Regarding coherent diagrams of the form $*\to C\rightrightarrows D$, they iso $\alpha :Fc\rightarrow Gc$ does not have to be natural in $c$, right? $\endgroup$ – Arrow Apr 19 '16 at 12:05

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