Today we will develop the theory of sheaves. We will simply ignore any set-theoretic issues.
The classical picture#
Let’s recall the classical situation. Let $X$ be a topological space. A presheaf $\mathcal{F}$ on $X$ is an assignment of a set $\mathcal{F}(U)$ for each open $U \subseteq X$, together with restriction maps $$ \rho_ {U,V} \colon \mathcal{F}(U) \to \mathcal{F}(V) $$ whenever $V \subseteq U$, which is functorial in the sense that $$ \rho_ {U,U} = \mathrm{id}, \quad \rho_ {V,W} \circ \rho_ {U,V} = \rho_ {U,W} $$ for $W \subseteq V \subseteq U$ of opens. One could also say that this is the same as a functor $$ \mathsf{Open}(X)^\mathrm{op} \to \mathsf{Set} $$ where $\mathsf{Open}(X)$ is the category with open subset of $X$ as objects and inclusions as morphisms.
A presheaf $\mathcal{F}$ is called a sheaf when it satisfies the sheaf condition. This says that if $U = \bigcup_i U_i$ is a covering, then $$ \mathcal{F}(U) \to \prod_i \mathcal{F}(U_i) \rightrightarrows \prod_ {i,j} \mathcal{F}(U_i \cap U_j) $$ is an equalizer.
What $\infty$-category theory teaches us is that $\mathcal{S}$ is better than $\mathsf{Set}$. We might imagine trying to replace $\mathsf{Set}$ by $\mathcal{S}$, so looking at the category of functors $$ \mathsf{Open}(X)^\mathrm{op} \to \mathcal{S} $$ instead. Then, we know what a limit diagram in $\mathcal{S}$ is, so maybe we can make sense of the sheaf condition. This is where we are headed.
The Yoneda embedding#
For presheaves, let’s work with a general $\infty$-category $\mathcal{C}$. In the $1$-categorical setting, we always have a functor $$ \mathcal{C} \to \mathsf{Fun}(\mathcal{C}^\mathrm{op}, \mathsf{Set}) $$ defined by sending $X$ to $\operatorname{Hom}(-, X)$. Since mapping spaces in $\infty$-category are spaces, we hope to get a similar functor $$ \mathcal{C} \to \mathsf{Fun}(\mathcal{C}^\mathrm{op}, \mathcal{S}) = \mathsf{PSh}(\mathcal{C}). $$ But one issue is that these mapping spaces are always up to homotopy, so it becomes a bit annoying to actually construct this.
The twisted arrow category#
Definition 1 (Kerodon 03JG). Let $\mathcal{C}$ be a simplicial set. The twisted arrow category $\operatorname{Tw}(\mathcal{C})$ is a simplicial set defined as sending $J \in \Delta$ to $$ \operatorname{Hom} _ {\mathsf{Set} _ \Delta}(J^\mathrm{op} \ast J, \mathcal{C}). $$
So objects are arrows, a morphism from $X \to Y$ to $Z \to W$ is a square $$ \begin{CD} Z @>>> X \br @VVV @VVV \br W @<<< Y \end{CD} $$ and so on.
There is a natural functor $$ \operatorname{Tw}(\mathcal{C}) \to \mathcal{C}^\mathrm{op} \times \mathcal{C} $$ defined by just remembering the $J^\mathrm{op}$ part or the $J$ part.
Proposition 2 (Kerodon 03JQ). Assume that $\mathcal{C}$ is an $\infty$-category. Then the map $\operatorname{Tw}(\mathcal{C}) \to \mathcal{C}^\mathrm{op} \times \mathcal{C}$ is a left fibration.
Proof.
We have a lifting problem to solve, and it can be solved.
Corollary 3 (Kerodon 03JR). If $\mathcal{C}$ is an $\infty$-category, then $\operatorname{Tw}(\mathcal{C})$ is also an $\infty$-category.
We haven’t talked about this, but a left fibration essentially is a functor from the base category to $\mathcal{S}$. For each point in the base, we can look at the fiber over that point, and it’s going to be a Kan complex. Then given a morphism, a lift will give you a map between the fibers. This works well in the $\infty$ setting as well, so we should think of this left fibration as a functor $$ \mathcal{C}^\mathrm{op} \times \mathcal{C} \to \mathcal{S}. $$ (For those who are familiar with Grothendieck fibrations of $1$-categories, this is the same thing.)
What does this functor look like? If we look at the fiber of this left fibration over $(X, Y)$, note that there is a natural inclusion $$ \operatorname{Hom} _ \mathcal{C}^\mathrm{L}(X, Y) = \mathcal{C} _ {/X} \times_ \mathcal{C} { Y } \hookrightarrow \operatorname{Tw}(\mathcal{C}) \times_ {\mathcal{C}^\mathrm{op} \times \mathcal{C}} {(X,Y)}. $$
Proposition 4 (Kerodon 03JW). The above inclusion is a homotopy equivalence of Kan complexes.
So we can really think of the functor $\mathcal{C}^\mathrm{op} \times \mathcal{C} \to \mathcal{S}$ as sending $(X, Y)$ to the mapping space $\operatorname{Hom} _ \mathcal{C}(X,Y)$.
Definition 5. We define the Yoneda embedding functor $$ h_ \bullet \colon \mathcal{C} \to \mathsf{Fun}(\mathcal{C}^\mathrm{op}, \mathcal{S}) $$ by adjunction to $\mathcal{C}^\mathrm{op} \times \mathcal{C} \to \mathcal{S}$.
The Yoneda lemma#
The Yoneda lemma, as we know it, holds in the context of $\infty$-categories.
Proposition 6 (Kerodon 03M3). Let $\mathcal{C}$ be an $\infty$-category, let $X \in \mathcal{C}$ be an object, and let $\mathcal{F} \colon \mathcal{C}^\mathrm{op} \to \mathcal{S}$ be a presheaf. Then the map $$ \operatorname{Hom} _ {\mathsf{PSh}(\mathcal{C})}(h_X, \mathcal{F}) \to \mathcal{F}(X) $$ induced by evaluation at $\mathrm{id} _X$ is an equivalence.
Corollary 7 (Kerodon 03M0). The Yoneda embedding $$ h_ \bullet \colon \mathcal{C} \to \mathsf{PSh}(\mathcal{C}) $$ is fully faithful.
Properties of the presheaf category#
Here are some other facts that might be useful.
Proposition 8. The category $\mathsf{PSh}(\mathcal{C})$ has all (small) limits and colimits, and they can be computed objectwise. (More precisely, a limit diagram evaluated at an object gives a limit diagram in $\mathcal{S}$.)
Proposition 9. The Yoneda embedding $h_ \bullet \colon \mathcal{C} \hookrightarrow \mathsf{PSh}(\mathcal{C})$ preserves all limits in $\mathcal{C}$.
Theorem 10 (Kerodon 03W9). Let $\mathcal{D}$ be a category which has all (small) colimits. Denote by $\mathsf{Fun}^\prime$ the full subcategory of the functor category that preserves all colimits. Then composition with $h_ \bullet$ induces an equivalence $$ \mathsf{Fun}^\prime(\mathsf{PSh}(\mathcal{C}), \mathcal{D})) \to \mathsf{Fun}(\mathcal{C}, \mathcal{D}) $$ of $\infty$-categories.
Informally, $\mathsf{PSh}(\mathcal{C})$ is the category obtained from $\mathcal{C}$ by “freely adjoining” all colimits.
Sheaves on a topological space#
Let $X$ be a topological space. We now restrict our attention to the $1$-category (even a $(0,1)$-category!) $$ \mathcal{C} = \mathsf{Open}(X) $$ of open subsets of $X$. We have a definition of presheaves on $X$, which is just $$ \mathsf{PSh}(X) = \mathsf{Fun}(\mathsf{Open}(X)^\mathrm{op}, \mathcal{S}). $$ We now want to put a sheaf condition, but what would this be?
The sheaf condition#
For each open cover $U = \bigcup_ {i \in I} U_i$, there is an associated sheaf condition on the presheaf $\mathcal{F} \in \mathsf{PSh}(X)$. We can formulate this condition in two different ways: using cosimplicial diagrams or using sieves.
The first one is what more people would be familiar with, but it turns out to be technically more challenging to use. Let $\Delta_+$ denote the category $\Delta$ together with the empty set $[-1] = \emptyset$, still with weakly increasing functions as morphisms. For each $[n] \in \Delta_+$, we can construct the space $$ C(U, {U_i}, \mathcal{F}) _ n = \prod_ {i_0, \dotsc, i_n \in I} \mathcal{F}(U_ {i_0} \cap \dotsb \cap U_ {i_n}) \in \mathcal{S}, $$ where when $n = -1$ this would just be $\mathcal{F}(U)$. Moreover, for any $[n] \to [m]$ in $\Delta_+$, we can define a map $A_n \to A_m$ by pulling back along appropriate restriction maps. At the end, we get a functor $$ C(U, {U_i}, \mathcal{F}) \colon N(\Delta_+) \to \mathcal{S}. $$ The first few terms will look like $$ \mathcal{F}(U) \to \prod_ {i} \mathcal{F}(U_i) \rightrightarrows \prod_ {i,j} \mathcal{F}(U_i \cap U_j) \to \dotsb. $$ Note that $\Delta_+$ can be thought of as the left cone on $\Delta$ since $[-1]$ is the initial object.
Definition 11 (Definition 1). A presheaf $\mathcal{F} \in \mathsf{PSh}(X)$ is a sheaf when the diagrams $C(U, {U_i}, \mathcal{F})$ are limit diagrams in $\mathcal{S}$ for all coverings $U = \bigcup_i U_i$.
Here is the simpler definition. Given an open covering $U = \bigcup_i U_i$, we can define a (classical) presheaf $$ \operatorname{Sieve} _ {U, {U_i}} \in \mathsf{PSh}(X); \quad V \mapsto \begin{cases} \ast & \text{if } V \subseteq U_i \text{ for some } i, \br \emptyset & \text{otherwise}. \end{cases} $$ This is naturally a sub-presheaf of $h_U \in \mathsf{PSh}(X)$ that sends $V$ to $\ast$ when $V$ is in $U$. So we have a natural map $$ \operatorname{Sieve} _ {U, {U_i}} \hookrightarrow h_U. $$
Definition 12 (HTT, Definition 6.2.2.6). A presheaf $\mathcal{F} \in \mathsf{PSh}(X)$ is a sheaf when the composition map $$ \mathcal{F}(U) = \operatorname{Hom} _ {\mathsf{PSh}(X)}(h_U, \mathcal{F}) \to \operatorname{Hom} _ {\mathsf{PSh}(X)}(\operatorname{Sieve} _ {U, {U_i}}, \mathcal{F}) $$ is a homotopy equivalence for all coverings $U = \bigcup_i U_i$.
Proposition 13. The two definitions are equivalent.
Proof.
The limit of the cosimplicial space $$ C(U, {U_i}, \mathcal{F}) \vert_ {\Delta} \colon \Delta \to \mathcal{S} $$ can be computed in the following way. First note that, using the Yoneda lemma, each term can be rewritten as $\operatorname{Hom} _ {\mathsf{PSh}(X)}(I_n, \mathcal{F})$, where $$ I_n = \coprod_ {i_0, \dotsc, i_n \in I} h_ {U_ {i_0} \cap \dotsb \cap U_ {i_n}}. $$ by the Yoneda lemma. Moreover, the cosimplicial maps are actually induced from maps between the presheaves $I_n$. Therefore we can write $$ \varprojlim C(U, {U_i}, \mathcal{F}) \vert_ \Delta = \operatorname{Hom} _ {\mathsf{PSh}(X)} \Bigl( {\textstyle \varinjlim_ {\Delta^\mathrm{op}}} I_ \bullet, \mathcal{F} \Bigr). $$ At this point, the colimit of $I_ \bullet$ can be computed objectwise, and this is a geometric realization because the indexing category of $\Delta^\mathrm{op}$. This explicit description identifies this colimit with $\mathrm{Sieve} _ {U,{U_i}}$. Therefore the limit condition becomes the condition that $$ \mathcal{F}(U) \to \operatorname{Hom} _ {\mathsf{PSh}(X)}(\mathrm{Sieve} _ {U,{U_i}}, \mathcal{F}) $$ is a homotopy equivalence.
Definition 14. We define $\mathsf{Sh}(X)$ as the full subcategory of $\mathsf{PSh}(X)$ consisting of sheaves.
Sheafification#
The natural inclusion $\mathsf{Sh}(X) \hookrightarrow \mathsf{PSh}(X)$ has a left adjoint $\mathsf{PSh}(X) \to \mathsf{Sh}(X)$ called sheafification.
Definition 15 (HTT, Construction 6.2.2.9). There is an operation $$ -^\dagger \colon \mathsf{PSh}(X) \to \mathsf{PSh}(X) $$ that informally can be written as $$ \mathcal{F}^\dagger(U) = \varinjlim_ {{U_i}} \varprojlim_ {\Delta} C(U, {U_i}, \mathcal{F}) _n. $$ There is moreover a natrual transformation from the identity functor to this functor, so that there is a map $\mathcal{F} \to \mathcal{F}^\dagger$ for every presheaf $\mathcal{F}$.
Lemma 16 (HTT, Remark 6.2.2.11). The functor $-^\dagger \colon \mathsf{PSh}(X) \to \mathsf{PSh}(X)$ commutes with all finite limits.
Lemma 17 (HTT, Lemma 6.2.2.14). If $\mathcal{F} \in \mathsf{PSh}(X)$ and $\mathcal{G} \in \mathsf{Sh}(X)$, then the natural map $$ \operatorname{Hom} _ {\mathsf{PSh}(X)}(\mathcal{F}^\dagger, \mathcal{G}) \to \operatorname{Hom} _ {\mathsf{PSh}(X)}(\mathcal{F}, \mathcal{G}) $$ is a homotopy equivalence.
Once we have this dagger functor, we can transfinitely apply this up to a large enough cardinal. This way, we may obtain the sheafification functor.
Theorem 18 (HTT, Proposition 6.2.2.7). The natural inclusion functor $\mathsf{Sh}(X) \hookrightarrow \mathsf{PSh}(X)$ has a left adjoint, called sheafification, and moreover the sheafification functor commutes with finite limits.
Examples of sheaves#
Here is the first class of examples.
Proposition 19. A classical presheaf (i.e., a presheaf valued in sets) is a sheaf in the classical sense if and only if it is a sheaf in the $\infty$-category sense.
Proof.
The point is that the natural inclusion functor $\mathsf{Set} \to \mathcal{S}$ is the right adjoint to $\pi_0 \colon \mathcal{S} \to \mathsf{Set}$, so that it commutes with all limits. This means that the sheaf condition (the limit version) can be checked in $\mathsf{Set}$. Now one can check that the “higher” conditions don’t contribute.
Here is a more interesting example.
Proposition 20 (HTT, Corollary 7.1.4.4). Assume $X$ is a paracompact space, and let $K$ be a Kan complex. Consider the constant presheaf $\underline{K} \in \mathsf{PSh}(X)$ that sends every open to $K$. The sheafification of presheaf is given by
For instance, if you look at the constant sheaf with value $K(A, n)$, the $\pi_0$ of the sections will precisely recover $H^n(U; A)$!
Poincaré duality#
This is a fun application of the theory of sheaves. Remember how you learned to prove Poincaré duality? First there was this duality for open balls, and then we had to try and glue these isomorphisms by finding covers of the manifold and doing a bunch of nasty Mayer–Vietoris comparisons when you take unions of these balls. This was so bad that you might have assumed the existence of a “good” cover.
The idea is that cohomology or homology is something like a sheaf, when viewed as a functor sending $U$ to the cohomology or homology of $U$. The idea is that there are local Poincaré isomorphisms, and by the sheaf property, this will glue to a global Poincaré isomorphism.
Usually the duality is stated as $$ H_ \ast(M; A) \cong H^{n-\ast}(M; A), $$ and these are covariant in the space. So instead of sheaves, we should be working with cosheaves.
Definition 21. The derived category $\mathsf{Mod} _ \mathbb{Z}$ (of cohomological chain complexes) has a simplicial enhancement, hence can be thought of as an $\infty$-category.
We will talk more about the derived category next time.
Definition 22. Let $\mathcal{C}$ be a category that has all limits. A sheaf on $M$ valued in $\mathcal{C}$ is a functor $\mathsf{Open}(M)^\mathrm{op} \to \mathcal{C}$ satisfying the sheaf conditions.
Now assume that $M$ is an $n$-manifold, which by definition is Hausdorff and second-countable and all that.
Proposition 23. A presheaf $\mathcal{F}$ on $M$ valued in $\mathcal{C}$ is a sheaf if and only if the following conditions are satisfied:
- the object $\mathcal{F}(\emptyset)$ is final,
- given $U, V \subseteq M$, the diagram $$ \begin{CD} \mathcal{F}(U \cup V) @>>> \mathcal{F}(U) \br @VVV @VVV \br \mathcal{F}(V) @>>> \mathcal{F}(U \cap V) \end{CD} $$ is Cartesian,
- for a countable increasing sequence $U_1 \subseteq U_2 \subseteq \dotsb$, the induced map $$ \mathcal{F}({\textstyle\bigcup} U_i) \to \varprojlim \mathcal{F}(U_i) $$ is an equivalence.
Proposition 24. If $\mathcal{F}$ is a sheaf on $M$ valued $\mathcal{C}$, then the natural map $$ \mathcal{F}(M) \to \varprojlim_ {U \in \mathfrak{B}} \mathcal{F}(U) $$ is an equivalence, where $\mathfrak{B} \in \mathsf{Open}(M)$ is the full subcategory of open subsets of $M$ homeomorphic to $\mathbb{R}^n$.
Now let us take $\mathcal{C} = \mathsf{Mod} _ \mathbb{Z}^\mathrm{op}$. Since homology and compactly supported cohomology are both covariant, we may consider them as presheaves $$ C_ {-\bullet}(-; A), C_c^\bullet(-; A) \in \mathsf{PSh}(M; \mathcal{C}) $$ valued in $\mathcal{C}$.
Proposition 25. Both presheaves are sheaves.
Proof.
We have Mayer–Vietoris that tells us that the diagram for $U, V, U \cap V, U \cup V$ is Cartesian. The other thing to verify is the fact about increasing sequences. Here we use the fact that filtered colimits in the derived category can actually be computed at the level of chain complexes.
At this point, we can compare $$ C_ {-\bullet}(M; A) \simeq \varinjlim_ {U \in \mathfrak{B}} C_ {-\bullet}(U; A) \simeq \varinjlim_ {U \in \mathfrak{B}} C_c^{n+\bullet}(U; A) \simeq C_c^{n+\bullet}(M; A) $$ in $\mathsf{Mod} _ \mathbb{Z}$ because we know that there is a canonical isomorphism $$ C_ {-\bullet}(U; A) \simeq C_c^{n+\bullet}(U; A) $$ when $U$ is homeomorphic to $\mathbb{R}^n$ and has an $A$-orientation. Taking cohomology groups of both chain complexes to recover Poincaré duality $$ H_i(U; A) \simeq H_c^{n-i}(U; A). $$