Schreiber heuristic introduction to sheaves, cohomology and higher stacks

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The following is (supposed to be) a pedagogical motivation of the concepts sheaf, stack, infinity-stack and higher topos theory. It assumes only that the reader has a heuristic knowledge of topological spaces and aims to provide from that a heuristic but useful idea of the relevance of the circle of ideas of categories and sheaves, nonabelian cohomology, sheaf cohomology and a bit of higher topos theory.

Motivation: maps between topological spaces

It is a familiar fact that classes of maps between topological spaces encode interesting information.

To pick just one simple but important example: there is a topological space that goes by the name $K(\mathbb{Z},2)$ or $\mathcal{B}U(1)$ and is called an Eilenberg-Mac Lane space. Whatever the name of this space, it is that space which has the peculiar property that for any other space $X$, the homotopy classes $[X, \mathcal{B}U(1)]$ of continuous maps from $X$ to this space $\mathcal{B}U(1)$ characterize the following useful information:

• they correspond to equivalence classes of circle bundle?s over $X$: that is, to spaces $P$ equipped with a map to $X$ such that over a contractible open subset $U \subset X$ the space $P$ looks like the product of $U$ with the circle, $P|_U \simeq U \times S^1$ (and such that the identification respects the action of the group $U(1)$ on the circle). One says that $\mathcal{B}U(1)$ is the classifying space for circle bundles: $[X, \mathcal{B}U(1)] \simeq \{$circle bundles over $X \}/{\sim}$.

  • one says that $\mathcal{B}U(1)$ is the classifying space for circle bundles or that it represents the assignment of circle bundles to spaces – to some extent the topic of sheaves, and cohomology is all about the notion of such representability

  • A famous such circle bundle you may have seen is the Hopf bundle, which is a way to map the 3-dimensional sphere $S^3$ to the familiar 2-dimensional sphere $S^2$, such that over small parts $U \subset S^2$ of the 2-sphere the 3-sphere looks like the product space $U \times S^1$.

  • This particular circle bundle corresponds to a very special class of maps $X \to \mathcal{B}U(1)$: namely the collection of maps $[S^2, \mathcal{B}U(1)]$ happens to have the structure of the abelian group $\mathbb{Z}$ of integers under addition, and the Hopf bundle corresponds to the the generating element $1$ of this group;

  • generally, this abelian group formed by $[X, \mathcal{B}U(1)]$ is famous under another name: it is called the second integral cohomology group of $X$, denoted $H^2(X, \mathbb{Z})$.

• There is a similar such cohomology group $H^{n+1}(X,\mathbb{Z})$ of $X$ for every natural number $n$: the $(n+1)$st integral cohomology group of $X$ – and all these groups are naturally realized in terms of homotopy classes of maps from $X$ into some topological space – called $K(\mathbb{Z}, n+1)$ or $\mathcal{B}^{n} U(1)$ – and again each element in these groups can be identified with a certain geometric structure living over $X$, called a circle $n$-bundle or $(n-1)$-gerbe: $[X, \mathcal{B}^{n} U(1)] = \{$circle $n$-bundles on $X \}/{\sim}$.

So these cohomology groups encode a lot of interesting information about topological spaces; but there is some useful information about a space, which is not encoded in maps from it to another topological space:

• in particular, if the space $X$ in question is not just a topological space but carries extra structure?, such as being a smooth manifold, then one is interested in smooth circle bundles over $X$ which are equipped with a connection $\nabla$: a good rule which assigns to each smooth path $\gamma : x \to y$ in $X$ an identification between the two circles over its endpoints;

  $$\array{ x && \stackrel{\nabla}{\mapsto} & S^1_x \\ \downarrow^\gamma &&& \downarrow^{\nabla(\gamma)} \\ y && \stackrel{\nabla}{\mapsto} & S^1_y }$$

  • this is the point where these considerations become of interest for physics?: there bundles with connection, and then $n$-bundles, or $(n-1)$-gerbes, with connection encode gauge fields such as the electromagnetic field.

  • you might guess that there is a smooth manifold classifying smooth circle bundles with connection – but there is not, not unless one stretches the meaning of manifold and of space in general a bit and come up maybe we just need a slightly more flexible notion of what “space” should mean.

• the generalization of the notion of space which does accomplish this: the notion of a sheaf

  • (there is a very general abstract nonsense way to understand sheaves as generalized spaces in the context of a very general abstract duality between the notions of space and quantity. The following is a heuristic way to understand this).

the basic idea of sheaves

To understand what a sheaf is, recall which aspect of the notion of topological space was relevant in the above examples $\mathcal{B}^{n} U(1)$: these spaces were entirely characterized by how one can map other spaces into them:

$$X \mapsto [X,\mathcal{B}^{n} U(1)] .$$

This is a general strategy that one can adopt: suppose I dream up a space but don’t tell you which one it is, but I give you hints: for each other space $U$ that you can dream up, I tell you how you can probe my space by mapping your space $U$ into it.

• Let’s now call the space which I dreamed up $X$, the generic symbol for spaces. For every space $U$ that you come up with, I do some secret computation and then present you with the result: I hand you a set, let’s call it $X(U)$, and tell you that this is the set of ways that $U$ can be mapped into $X$:

  $X(U) = \{$ways to map $U$ into some hypothetical $X \}$

    =: {probes of $X$ by $U$}.

• will you be able, in general, to guess my space $X$ from this information $U \mapsto X(U)$? No. So, to be fair, I should provide a bit more information: what you need to know to actually get a feeling for what my space $X$ is like is an idea about how different ways of probing my space relate to each other.

• Okay, so we’ll do this: for every pair $U$ and $V$ of test spaces that you come up with, and for every way $\phi : U \to V$ of mapping these into each other, I inform you not only about the set of ways $X(U) = \{U$-probes of $X\}$ that $U$ can be mapped into my secret space $X$, and the set of ways $X(V) = \{V$-probes of $X\}$ that $V$ can be mapped into my secret space, but I also tell you how these are related when you first map $V$ into $X$ by a map $(p : V \to X) \in X(V)$ and then map $U$ into $X$ by first mapping it to $V$: $U \stackrel{f}{\to} V \stackrel{p}{\to} X$.

• This transforms every element in $X(V)$ into an element in $X(U)$. To be fair, I should tell you at least what this transformation is!

• So I’ll do that: for every map of test spaces $f : U \to V$ that you hand me, I return you a map of sets that I denote $X(\phi) : X(V) \to X(U)$, which indicates how $V$-probes of $X$ turn into $U$-probes when you precompose them with $\phi : U \to V$.

Do you need still more information to guess my space $X$? It turns out that: no, this information is enough!

• This somewhat remarkable fact is closely related to a fundamental statement of category theory: the Yoneda lemma and its implication on representable functors. These terms will be discussed in a moment…

First, there is a bit more to the game than mentioned so far: you need to be sure that even if I won’t reveal my space $X$ to you directly, the little information about it which I do provide, that rule $U \mapsto X(U)$, I should provide honestly and consistently. Some consistency checks to assure that I am not just making things up but am giving you consistent information about my secret space $X$ are the following:

• it must be true that when you hand me the identity map $Id_U : U \to U$ on a test space, that then then I return you the identity map $Id_{X(U)} : X(U) \to X(U)$ on the set of probes of $X$ by $U$;

• also, it must be true that when you first hand me two consecutive maps of test spaces, $f : U \to V$ and $g : V \to W$, and then hand me their composite $g \circ f : U \to W$, that my reply $X(g \circ f) : X(W) \to X(U)$ to the latter is the result of composing my two replies about the former: $X(g \circ f) = X(W) \stackrel{X(g)}{\to} X(V) \stackrel{X(f)}{\to} X(U)$.

Such an assignment $U \mapsto X(U)$ of sets to spaces which satisfies the above two properties is what is called a presheaf: this, in turn, is nothing but what is called a functor on a category:

• the category in question here is the collection of test topological spaces $U$, called objects, equipped with continuous maps $f : U \to V$ between them, called morphisms, which may be composed when their domains and targets match. This composition is associative in the obvious way and on every object $U$ there is an identity map $Id_U : U \to U$, composition with which does nothing. This category is known by the name Top; $Top = \{$topological spaces and continuous maps between them$\}$

• the other category in the game, namely the category called Set whose objects are sets and whose morphism?s are functions between sets: $Set = \{$sets and functions between them$\}$

• the probe-game which we are playing can hence be understood as being an assignment of object?s in Set? to objects in Top and of morphisms in Set to morphisms in Top which satisfies the above consistency conditions: this is called a contravariant Set-valued functor, or a presheaf, on Top and it is denoted $X(-) : Top^{op} \to Set$

We say presheaf? instead of just functor?, even though taken at face value a presheaf is just a functor, to indicate that there is one further, more nontrivial consistency check on the information about probes of my secret space $X$ which you should do: it must be true that you can piece together the information which I give for small test spaces to deduce information about probes by bigger test spaces.

• More precisely, if I tell you the set $X(U)$ of probes of my secret space $X$ by the test space $U$, and if you then chop up $U$ into two pieces $V_1$ and $V_2$ sitting inside $U$ by inclusion maps $p_i : V_i \hookrightarrow U$, with a bit of overlap $V_1 \cap V_2$, then it ought to be true that all the probes by $U$ in $X(U)$ can be entirely and exactly be reconstructed from taking probes by $V_1$ in $X(V_1)$ and by $V_2$ in $X(V_2)$ and see if they match over the overlap $V_1 \cap V_2$ of the two small probes:

• in symbols, the collection of pairs in $\{$matching maps from $V_1$ and $V_2$ to $X \} \hookrightarrow X(V_1) \times X(V_2)$ whose elements coincide when restricted to the overlap

  $$X(V_1) \times X(V_2) \stackrel{\stackrel{restrict first element}{\to}}{\stackrel{restrict second element}{\to}} X(V_1 \cap V_2)$$

  should be exactly all the possible maps in $X(U)$: $\{$matching probes of $X$ on $V_1$ and $V_2 \} \simeq X(U)$.

• one says that the set $\{$matching probes of $X$ on $V_1$ and $V_2 \}$ is the equalizer? of the two parallel morphisms?

  $$X(V_1) \times X(V_2) \stackrel{\stackrel{restrict first element}{\to}}{\stackrel{restrict second element}{\to}} X(V_1 \cap V_2) :$$

  the optimal solution to mapping into $X(V_1) \times X(V_2)$ such that these two maps become equal;

• this condition is called a sheaf? or descent? condition: one thinks of the sheaf of descending from the cover to the base along the map $(V_1 \sqcup V_2) \stackrel{p_1 \sqcup p_2}{\to} U$;

This now is a sensible game to play: I don’t tell you directly which space $X$ I am thinking of, but I do give you all the information $U \mapsto X(U)$ about its probes $X(U)$ by test spaces $U$, subject to the conditions that

• $X(-)$ is functorial and hence a presheaf?;

• $X(-)$ satisfies descent? and is therefore a sheaf?

sheaves more general than spaces

So far we have been talking about using topological spaces as probe spaces, hence about presheaves on the category Top?. One can use other systems $S$ of test spaces as long as $S$ forms a category? and such that there is an agreement about how some of its object?s $U$, can be cover?ed by other object?s, such as the $V_1$ and $V_2$ above. Such an agreement of what counts as a cover? in a category? is called a coverage? or a Grothendieck topology?, and a category? equipped with such an information is called a site?;

Here are some important examples:

• A popular choice is to fix one particular topological space $Z$ and probe all generalized spaces using only open subsets $U \subset Z$ of $Z$. These open subsets form the category of open subsets? $S = Op(Z)$ and the choice of coverage? is the obvious one. Sheaves? on $Op(Z)$ are generalized spaces over $Z$.

• Another relevant choice is to use instead of all topological test spaces in the category Top? all smooth manifolds as test spaces, in the category $S =$ Diff?.

• Or the combination of these: the test spaces in the category $Op(Z)$ for $Z$ a smooth manifold.

We had motivated this language of sheaves as a reformulation of the familar notion of spaces in a somewhat more indirect form: a rule for how to probe a space. But now something strange happens: it turns out that there are collections of probe information $X(-) : S^{op} \to Set$ which do obey the rules, i.e. do form a sheaf, but for which there is nevertheless no test space in $S$ that it describes probes into. And in fact, this is the generic case.

• On the other hand, maybe this just means that our notion of “ordinary space” was a bit too restrictive: after all, we noticed that all that really matters usually are the ways we can probe a given space by other spaces; and that the information about such probes is useful if only it obeys the rules of being a sheaf.

• So we have every justification then to regard sheaves in general as perfectly valid generalized spaces; or more precisely: we regard sheaves as rules for how to probe generalized spaces, and we take these generalized spaces to be entirely specified by their sheaves? of probes.

Examples:

• The notion of “generalized” here depends on the perspective of our probes. For the probe category of the form $S = Op(Z)$ with $Z$ a topological space, most sheaves are “generalized spaces” in that they do not describe maps into a space in $S$, i.e. an open subset of $Z$. But every sheaf on $Op(Z)$ can be shown to correspond to maps from $X$ to an ordinary topological space $E$ which is equipped with a continuous map $p : E \to X$ down to $X$, such that $p$ is locally a homeomorphism. Such $E$ are called étale spaces? over $X$.

• A sheaf on $S =$ Diff? is a generalized smooth space?. These may be very different from ordinary manifolds and ordinary topological spaces. There are generalized smooth spaces with a single point and still many curves in them. In fact, the generalized classifying space for smooth circle bundles with connection that we are still after is of this form.

maps between generalized spaces

Since we are thinking of sheaves $F_X : S^{op} \to Set$ and $F_Y : S^{op} \to Set$ as characterizing ‘generalized spaces’ $X$ and $Y$, we better have a good notion of maps between sheaves $F_X \to F_Y$ that corresponds to a sensible notion of maps $X \to Y$ between generalized spaces.

That’s easy to figure out: if $f : X \to Y$ is supposed to be a map between generalized spaces, whatever these are, then in particular it turns every probe $p \in X(U)$, of $X$ which we are thinking of as a probe map $p : U \to X$, to a probe $f \circ p \in Y(U)$ of $Y$, which we should think of as the composite map $f \circ p : U \stackrel{p}{\to} X \stackrel{f}{\to} Y$ that simply maps the image of $U$ in $X$ further to an image of $U$ in $Y$ using the map $f : X \to Y$.

• nota bene: for the time being there is no space $X$ such that thinking of probes $p \in X(U)$ as maps $p : U \to X$ of some sort would literally make sense. However, once we impose some consistency conditions on the probe sets $X(U)$ precisely this will make sense. And to motivate these consistency conditions in the first place, it is best to already think of probes $p \in X(U)$ as being maps $p : U \to X$.

• so in terms of their collections (sheaves) of probes a map $f : X \to Y$ of generalized spaces should be given for each probe space $U$ by a map of sets of probes $f_U : X(U) \to Y(U)$, to be thought of as encoding the post-composition map $f_U : (U \stackrel{p}{\to} X) \mapsto (U \stackrel{p}{\to} X \stackrel{f}{\to} Y)$

• but we can also pre-compose: given a map $\phi : U \to V$ of test spaces, we also have the probe transformation map from above, $X(\phi) : X(V) \to X(U)$ which is to be thought of as acting by $X(\phi) : (V \stackrel{p}{\to} X) \mapsto (U \stackrel{\phi}{\to} V \stackrel{p}{\to} X)$

• and we can combine both pre- and postcomposition and send $(V \stackrel{p}{\to} X) \mapsto (U \stackrel{\phi}{\to} V \stackrel{p}{\to} X \stackrel{f}{\to} Y)$

• in this notation it is manifestly clear that it must not matter whether we first pre- and then post-compose, or the other way around. But recall that so far writing $p : V \to X$ for a probe $p \in X(V)$ is just notation. So we need to enforce the obvious here on our formalism explicitly, lest the interpretation of generalized spaces breaks down.

• namely this means that the map $X(V) \stackrel{f_V}{\to} Y(V) \stackrel{\phi^*}{\to} Y(U)$ which first turns a $V$-probe of $X$ into a $V$-probe of $Y$ and then turns the result into a $U$-probe of $Y$, must coincide with the map $X(V) \stackrel{\phi^*}{\to} X(U) \stackrel{f_U}{\to} Y(U)$ which first turns a $V$-probe of $X$ into a $U$-probe of $X$ and then turns the result into a $U$-probe of $Y$.

• Such an equality between two different composite maps is conveniently depicted as a square

  $$\array{ X(U) &\stackrel{f_U}{\to}& Y(U) \\ \uparrow^{X(\phi)} && \uparrow^{Y(\phi)} \\ X(V) &\stackrel{f_V}{\to}& Y(V) }$$

  The fact that the two ways of going from the bottom left to the top right of the square are supposed to be equal is called. “the square commutes”, or “the square is commutative”.

So that then is our definition of maps $f : X \to Y$ between generalized spaces $X$ and $Y$: if $U \mapsto X(U)$ and $V \mapsto X(V)$ are the probe-assignments that define $X$ and $Y$, then the map $f$ is defined by probe-transformations $f_U : X(U) \to Y(U)$ such that for all maps $\phi : U \to V$ the above squares commute.

• Such a collection of data is called a natural transformation? between the probe-assigning functor?s $X(-)$ and $Y(-)$. Clearly, its a transformation, and it being “natural” just means that it satisfies the obvious consistency condition we discussed.

With this understanding of maps between generalized spaces in hand, we should go back and compare it to the notion of probes of generalized spaces by ordinary spaces.

• we had started the discussion with observing that every ordinary topological space $X$ defines a sheaf? of probes by the assignment $U \mapsto X(U)$ := {maps from $U$ to $X$} =: $Maps(U,X)$.

• this way every ordinary space is a generalized space, of course;

• but that should make us nervous: because it means that now there are two different definitions of what a map from a test topological space $U$ to a topological space should be:

  • on the one hand, it should be an element in $X(U) = Top(U,X)$, where $Top$ indicates a set of maps of topological spaces;

  • on the other hand, we can regard both $U$ and $X$ as generalized spaces and then look at the set $Sh(U,X)$ of maps of generalized spaces between them, as just discussed above;

  • if these two notions of maps don’t coincide, then the whole picture of generalized spaces so far runs into inconsistencies!

  • luckily it doesn’t, as one can check. Both notions of maps, while defined differently, happen to be perfectly equivalent. This statement is a simple exercise to prove, and still is the fundamental fact which makes the important idea of generalized spaces works – this fact is called the Yoneda lemma?: it says that

    $$\begin{aligned} Top(U,X) &=: X(U) \\ & = Sh(U,X) \end{aligned}$$

    or more formally

    $$Hom_{Sheaves}(U,X) \simeq X(U) .$$

• as a slogan: The Yoneda lemma says that the idea of generalized spaces determined by their collections of probes is consistent.

This point is so important that it is worthwhile to build it into our very notation, so that we can’t help but think of the situation in this way.

• We agree for any two generalized spaces $X$ and $Y$ to write $X(Y)$ for the collection of maps from $Y$ to $X$: $X(Y) := Sh(Y,X)$, the set of maps of generalized spaces from $Y$ to $X$.

• If $Y = U$ happens to be an ordinary space, then $X(U)$ consistently and equivalently denotes both: the collection of probes of $X$ by the ordinary space $U$, and the collection of maps of $U$ into $X$, with $U$ regarded as a generalized space itself.

• (This notation is for instance used in the book Categories and Sheaves?, see page 25.)

and more: a topos of generalized spaces

It turns out that we can keep going this way. There is not just a natural notion of maps between generalized spaces given by sheaves?, but every important type of operation on spaces has its analogs as an operation on sheaves: one says that sheaves form a topos?: a place where we can go to study generalized homotopy theory?, more general than the place of topological space?s, but where still all the crucial constructions familiar from topological spaces makes sense.

higher sheaves: $\infty$-stacks

Once we are at this point, we should go even a bit further: we have been talking about sets of probes $X(U)$. But of course for $X$ a true topological space? and $U$ a topological space?, there is not just a set, but a topological space? of maps $U \to X$ (using the compact-open topology on spaces of continuous maps). So we should be looking at sheaves with values in topological spaces: functors $X(-) : S^{op} \to Top$ satisfying some suitable condition.

There is only one thing we need to take care of, then: since a topological spaces is a bit more flexible than a plain set, we should be a bit careful about when two sheaves $U \mapsto X(U)$ with values in topological spaces are to be regarded as equal for all practical purposes: recall that $X(U)$ is to be thought of as the space of all ways of mapping a probe space $U$ into a space $X$. It’s not the single points (maps) in this space which matter so much, but they way they hang together.

• All “ways of hanging together” in a topological space $X(U)$ are measured by a collection of groups called the homotopy group?s $\pi_n(X(U))$, one for each natural number $n$.

• So if we are being careful, we should recognize that two sheaves $X : S^{op} \to Top$ and $Y : S^{op} \to Top$ must characterize essentially the same generalized space if there is a rule $f : X \to Y$ for turning for each test space $U$, probes of $X$ into probes of $Y$, $f_U : X(U) \to Y(U)$, as before, such that for sufficiently small $U$ this comparison map of probe spaces does not change the homotopy groups, i.e. so that all induced homomorphism $\pi_n(f_U, x) : \pi_n(X(U)) \to \pi_n(X(V), f_U(x))$ of homotopy group?s are isomorphisms of groups. One says that such $f_U$ is a weak homotopy equivalence?.

• If a comparison map $f : X \to Y$ exists with these properties (for small enough $U$, recall), then one says that our generalized spaces $X$ and $Y$ defined by these sheaves are weakly equivalent or quasi-isomorphic?.

There are several ways to deal with this complication that sheaves with values in topological spaces are very flexible, so flexible that it may require a bit of effort to determine if two of them determine the same generalized space, for all practical purposes.

One nice way to deal with it is to again go back to our attitude that everything should be determined by how we map other things into it: if $f : X \to Y$ is a weak equivalence of sheaves, as above, then we we would expect that for $Z$ any other sheaf, the induced map of spaces of probes $Z(f) : Z(Y) \to Z(X)$ of $Z$ by $X$ and $Y$ is a weak homotopy equivalence?.

• Recall from the discussion in the above section “maps of generalized spaces” that $Z(Y)$ is just another way saying “space of maps of generalized spaces from $Y$ to $Z$”, and similarly for $Z(X)$.

Now, that may fail to be true, simply because we are using an awkward realization of the sheaf $Z$. It may be that we have to straighten out $Z$ a bit, without really changing it, for the above to be true. Indeed, if things are set up correctly, there should always a sheaf of topological spaces $Z'$ which is equivalent to $Z$, but for which the above consistency condition is true: the map $Z'(f) : Z'(Y) \to Z'(X)$ between spaces of probes of $Z'$ by $X$ and by $Y$ is a homotopy equivalence.

The sheaves with this niceness property are called infinity-stack?s or (infinity,1)-sheaves?. Recall that despite the fancy terminology, these things are nothing but consistent rules for something that may be probed with test spaces $U$, $V$, …

• in applications this means that before we compute the space of maps $[X,A]$ between generalized spaces, we first have two ensure that at least one of them is sufficiently straightened out. Once says one picks a resolution. More on that below.

generalized cohomology

We come back to our original motivation. Recall that we were looking at cohomology groups of spaces $X$ by mapping $X$ into other spaces. We noticed that there is in general interesting information about $X$ which cannot be gained by mapping it into an ordinary topological space. Starting from this we have now arrived at a more general notion of space, called a higher sheaf or an $\infty$-stack.

• So let’s state for the record that we succeeded: we will say now for $X$ and $A$ generalized spaces, $\infty$-stacks, that the collection of maps

  $$H(X,A) := [X,A]$$

  from $X$ to $A$ is the cohomology? of $X$ with coefficients in $A$.

• Indeed, one finds that despite their enlarged generality, these $\infty$-stacks behave in all relevant aspects just like ordinary sheaves and ordinary topological spaces do. So one says that $\infty$-stacks form an infinity-topos? (more precisely: an (infinity,1)-topos?): a place where we can go in order to do the familiar homotopy theory? of topological spaces in a more general context.

complexes of sheaves: abelian $\infty$-stacks

Another term for this generalized cohomology? theory thus obtained is nonabelian cohomology?. The reason for that terms is a historic one: before arriving at the full picture of higher topos theory as described here, people had a pretty good guess about some aspects of this story, and this aspect they called sheaf cohomology?. It turns out that sheaf cohomology is precisely the abelian part of general cohomology. So now general cohomology is sometimes called nonabelian cohomology? to distinguish it from the more traditional cohomology theory.

Here is what “abelian part of general cohomology” means:

• For many sheaves that appear in practice, the topological space $X(U)$ assigned to any test space $U$ is not just a topological space, but happens to have the structure of an abelian group, too.

• In this case a cascade of simplifications kicks in: first of all, such topological spaces, by a special case of something called the homotopy hypothesis? (which is now really a precise theorem) can be identified with simplicial abelian groups.

• Second, by another theorem called the Dold-Kan correspondence?, simplicial abelian groups are equivalently encoded more simply in the collection of data given by (non-positively graded) chain complexes in abelian groups.

• {abelian chain complexes} $\simeq$ {simplicial abelian groups} $\subset$ {Kan complexes} $\simeq$ {nice topological spaces}

By an abelian sheaf? one means a sheaf $U \mapsto X(U)$ which takes values in chain complex?es of abelian groups (or, more generally, in complex?es in another abelian category?). Under the above correspondences, we understand this as a special case of a sheaf with values in topological spaces.

• By abelian sheaf cohomology? one means nothing but the above general higher topos theoretic? notion of cohomology given by the simple idea of collections of maps from one generalized space to another. Only that in the special case of abelian sheaves a wealth of particular computational techniques are used to compute these mapping spaces (such as derived categories? and derived functors?) which are not available in the fully general case. This wealth of computational machinery employed, useful as it is, sometimes hides the simple conceptual reasoning underlying everything.

There is a collection of tools available for computing explicitly the cohomologies $H(X,A) := [X,A]$ for the case that the coefficient space can be thought of as a complex of abelian groups, or a complex of modules, these tools go by names like

• model category?

  • derived category?

  • multiplicative system? / factorization system?

  • localization?

• derived functor?

  • injective resoltution?

  • bar and cobar construction?

and so forth; historically many concepts in abelian sheaf theory were thought to be defined in terms of these constructions. From a more modern perspective all these are just tools for unraveling an independent reality. This also explains why there are so many different tools which yield the same results. Most notably in the context of abelian sheaf cohomology? there are two general ways for computing this cohomology called

• right derived global section functor

• Čech cohomology;

these two methods differ in whether one computes using “resolutions” of either the coefficient object $A$ (this happens in the right derived global section functor approach) or in the domain object $X$ (this happens in Čech cohomology).

Examples

Recall that we started the search for a generalized notion of space motivated by the fact that a circle bundle on a space $X$ may be encoded by a map from $X$ to some topological space $\mathcal{B}U(1)$, but that the same was not true for circle bundles with connection. We had wanted to obtain a generalized notion of space such that this is remedied.

• Indeed, this is now the case. There is a famous abelian sheaf, called the second Deligne complex? which we may denote by $[P_1(-), \mathcal{B}U(1)]$ or $\bar \mathcal{B}_1 U(1)$ (as explained at Deligne cohomology?), which has precisely the property that maps from a manifold $X$ into it classify circle bundles with connection on $X$. Moreover, by varying the two parameters here we get the following cases

  • $A = [P_0(-), \mathcal{B}U(1)] = \mathcal{B}U(1)$; $H(X,A) = H^2(X,A)$ is second integral cohomology – classifies line bundles;

  • $A = [P_1(-), \mathcal{B}U(1)] = \bar \mathcal{B}_1 U(1)$; $H(X,A) = \bar H^2(X,\mathbb{Z})$ is second differential integral cohomology – classifies line bundles with connection;

  • $A = [P_2(-), \mathcal{B}U(1)] = \bar \mathcal{B}_2 U(1)$; $H(X,A) = \bar H_{flat}^2(X,\mathbb{Z})$ is second flat differential integral cohomology – classifies smooth line bundles with flat connection;

  • $A = [P_0(-), \mathcal{B}^2U(1)] = \mathcal{B}^2U(1)$; $H(X,A) = H^3(X,A)$ is third integral cohomology – classifies line bundle gerbes;

  • $A = [P_1(-), \mathcal{B}^2U(1)] = \bar \mathcal{B}^2_1 U(1)$ – classifies line bundle gerbes with connective structure;

  • $A = [P_2(-), \mathcal{B}^2U(1)] = \bar \mathcal{B}^2_2 U(1)$; $H(X,A) = \bar H_{flat}^3(X,\mathbb{Z})$ is third differential integral cohomology – classifies smooth line bundle gerbes with connection;

  • $A = [P_3(-), \mathcal{B}^2U(1)] = \bar \mathcal{B}^2_3 U(1)$; $H(X,A) = \bar H_{flat}^3(X,\mathbb{Z})$ is third flat differential integral cohomology – classifies line bundle gerbes with flat connection;

Notice that all these objects $A$ do indeed behave, and can be treated like, generalized spaces. For instance one can look at paths in these spaces and for instance form their fundamental groupoid?s. One finds results such that for instance

• $\Pi_1(\bar \mathcal{B}_2 U(1)) \simeq \mathcal{B}R$

• more generally: for $G$ any Lie group, we have $\Pi_1(\bar \mathcal{B}_2 G) \simeq \mathcal{B}\hat G$, where $\hat G$ is the unique simply connected Lie group covering $G$.