Hussain Kadhem
2022-09-13
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Let \(C_\bullet, C'_\bullet\) be chain complexes. A chain morphism \(f_\bullet: C_\bullet \to C'_\bullet\) of homogeneous degree \(d\) is a collection of maps \(f_n: C_n \to C'_{n+d}\). The collection of these maps can themselves be organized into a chain complex: the mapping complex \([C_\bullet, C'_\bullet]_\bullet\), where the objects of degree \(d\) are the chain morphisms of homogeneous degree \(d\), and the boundaries are given by \(\partial f = (-1)^d \partial \circ f - f \circ \partial\).
Let \(C\) be a \(\operatorname{DG}\)-category. The \(\operatorname{DG}\) nerve of \(C\) is a \(\infty\)-category with simplices constructed as follows:
The \(n\)-simplices are collections of the following data:
A collection \(x := x_{0 \leq i \leq n}\) of objects in \(C\);
for each subset \(x_I := x_{i_0<\cdots<i_k}\) of \(k \geq 2\) elements of \(x\), an element \(f_I\) of \([x_{i_0}, x_{i_k}]_{k-1}\)
such that the \(f_I\) satisfy a compatibility condition for composition witnessed by the homology: let \(f_I^{\leq j} = f_{i_0, \cdots, i_j}\), \(f_I^{\geq j} = f_{i_j, \cdots, i_k}\), and \(f_I^{\hat j} := f_{I \setminus j}\). Then they must satisfy: \[\partial f_I = \sum_{j=1}^{k-1} (-1)^j f_I^{\geq j} \circ f_I^{\leq j} - f_I^{\hat j}.\]
A site \((C,J)\) is a category \(C\) equipped with a structure \(J\) called a coverage.
A coverage is the assignment, to each object \(U\) in \(C\), a collection of covering families \(\{U_i \to^{f_i} U\}\), which are stable under pullback to a different object \(U'\).
A sheaf on a (nice) site \((C,J)\) is a presheaf \(X\) on \(C\), such that the following diagram is an equalizer: \[X(U) \to \prod_i X(U_i) \rightrightarrows \prod_{j,k} X(U_j\times_U U_k).\]
We can restate this in the language of sieves. A sieve \(S\) on an object \(U\) is a subfunctor of the Yoneda embedding \(Y_U := C(-, U)\). We think of \(S\) as distinguishing a set of morphisms into \(U\) which are stable under inclusion. A covering family \(\{U_i \to^{f_i} U\}\) generates a sieve by sending \(V\) to the subset of morphisms \(V \to U\) which factor through some \(f_i\).
Formally this is given by a coequalizer (in \(\operatorname{PSh}(C)\)) of the diagram \[\coprod_{i,j} Y_{U_i} \times_U Y_{U_j} \rightrightarrows \coprod_i Y_{U_i}.\] (The pullbacks are just given by intersections.)
A site on \(C\) can also be given by an assignment \(J(c)\) of distinguished families of sieves to every object \(c\). These are required to be stable under pullback: for every \(S \in J(c)\) and \(f: c' \to c\), the sieve \(f^*S \in J(c')\). They are also required to satisfy a locality condition, which basically says that if the union of the restrictions of a sieve \(S\) on \(c\) contains a sieve in \(J(c)\), then it must also be in \(J(c)\).
A sheaf on a (nice) site \((C,J)\) is a presheaf \(X\) on \(C\), such that the following diagram is an equalizer: \[X(U) \to \prod_i X(U_i) \rightrightarrows \prod_{i,j} X(U_i\times_U U_j).\]
In terms of sieves, this can be restated as the following diagram being an equalizer: \[[S(\{U_i\}), X] \to \prod_i X(U_i) \rightrightarrows \prod_{i,j} X(U_i \cap U_j).\]
Consider the category \(\operatorname{Sh}(C)\) of sheaves on \(C\). it enjoys some nice categorical properties: it has all finite limits, a closed Cartesian structure, and a subobject classifier. A category with these properties is called a topos, and we think of it as something that behaves like a category of sheaves on a site.
Let \(C,C'\) be \(\infty\)-categories. A \(\infty\)-functor \(C \to^F C'\) is a natural transformation between them, thought of as \(\Delta\)-presheaves.
It is an important feature of \(\infty\)-functors that they are not strictly compatible with composition of morphisms. If \(f,g\) are composable morphisms, or \(1\)-simplices, in \(C\), then it is not necessary that \(F(g)\circ F(f) = F(g \circ f)\). Instead, they will have a coherence relation witnessed by higher-dimensional morphisms.
For ordinary categories \(C,C'\), there is a natural functor category \([C,C']\) having functors as objects, and natural transformations as morphisms.
We would like to extend this construction in some natural way to the \(\infty\) setting. That is, construct a \(\infty\) category structure for \([C,C']\).
As a simplicial set \(F_\bullet\), this would look like a map \(F([n]) \mapsto [C, C'].\) However there is also a natural identification, given by the Yoneda embedding: \(F([n]) \cong [\delta^n, [C, C']]\). The right-hand side of this suggests a natural way of defining \(F_\bullet\) by applying the product-hom adjunction: \[F([n]) = [\delta^n \times C, C'].\]
This is indeed the right simplicial set structure for defining the \(\infty\)-category of \(\infty\)-functors from \(C\) to \(C'\). In fact, this can be extended to an \(\infty\)-category of \(\infty\)-categories.
Exercise. Work out the evaluation map \(\operatorname{Ev}: [C, C'] \times C \mapsto C'\).
Exercise. Recall the nerve \(N_\bullet(C)\) of an ordinary category, and show that there is an isomorphism of simplicial sets \[N_\bullet([C, C']) \cong [N_\bullet(C), N_\bullet(C')].\]
There is an isomorphism \(\operatorname{op}\) on \(\Delta\) which sends objects \([n] \mapsto [n]\), and reflects morphisms using the order-reversing function \(i \mapsto n-i\).
Let \(C\) be an \(\infty\)-category. Its opposite \(\infty\)-category \(C^{\operatorname{op}}\) is given by precomposition with \(\operatorname{op}\).
Let \(S_\bullet\) be a simplicial set which satisfies the Kan lifting condition for both inner and outer horns. Then it is a model of a \(\infty\)-groupoid: a \(\infty\)-category where all morphisms are (weakly) invertable. The collection of all of these form a \(\infty\)-category, which we denote \(\operatorname{sSet}\).
Let \(C\) be an \(\infty\)-category. Then an \(\infty\)-presheaf on \(C\) is an \(\infty\)-functor \(C^{\operatorname{op}} \to \operatorname{sSet}\). There is a corresponding \(\infty\)-category of \(\infty\)-presheaves \([C^{\operatorname{op}}, \operatorname{sSet}]\). This enjoys the same free cocompletion properties as for ordinary categories, coming from a universal \(\infty\)-Yoneda embedding.
A \(\infty\)-site is an \(\infty\)-category \(C\) equipped with a collection \(J(c)\) of sieves for each object \(c \in C\), which are defined to be sub-\(\infty\)-functors of the \(\infty\)-Yoneda-embedding \(Y_c\). These are required to satisfy the same properties as in the ordinary case. This is actually not as broad a generalization as it seems: there is a bijection between \(\infty\)-sites on \(C\) and ordinary sites on its homotopy category \(HC\).
An \(\infty\)-sheaf \(X\) on the site \((C,J)\) is a presheaf which is required to satisfy the descent condition: for all objects \(c\) and sieves \(S \hookrightarrow Y_c\), the following morphism is a weak equivalence: \[X(U) \cong [Y_U, X] \rightarrow [S, X].\]
We call a \(\infty\)-sheaf on a \(\infty\)-site a derived stack.
Example. The case we will care most about is that of derived stacks on the site of the opposite of simplicial commutative rings, equipped with the Zariski topology. (This is the site of derived affine schemes.)
Let \(X\) be a derived stack. The \(\infty\)-category of ind-coherent sheaves on \(X\) will be constructed as the \(\infty\)-category of ind objects of the category of coherent sheaves on \(X\). These are of course \(\infty\)-sheaves, for which we haven’t defined the notion of coherent. (We will defer this to a later time.) The takeaway is that this category enjoys a richer categorical structure with respect to some important functors. To explore this, we will take a detour through some topology.
Let \(X,Y\) be topological spaces and \(f: X \to Y\) a continuous map. Denote by \(\mathcal{O}(X),\mathcal{O}(Y)\) the categories of open subsets and inclusions. These form sites which we denote \(S_X,S_Y\).
Denote by \(T_\bullet\) the topos of sheaves on \(S_\bullet\). Objects are given by local homeomorphisms to the underlying topological space \(\bullet\).
The map \(f\) induces an inverse functor on open sets \(f^{-1}: \mathcal{O}(Y) \mapsto \mathcal{O}(X)\). This, in turn, induces a pullback functor on sheaves \(f_*: T_X \mapsto T_Y\), which is the direct image functor associated to \(f\).
On the other hand, from topology we have for any local homeomorphism \(E \to Y\) the pullback along \(f\): \(f^*E \to X\). This induces a functor \(f^*: T_Y \to T_X\), which is the inverse image functor associated to \(f\).
The functor \(f^*\) is left-adjoint to \(f_*\). (The directions here are very important.)
To see how \(f^*\) is naturally a left-adjoint, we recall that left adjoint functors are required to preserve colimits. Suppose that \(E \to^\pi Y \in T_Y\) is a local homeomorphism which is a colimit of a diagram \(J\). Consider the pullback square for \(E\): \[\begin{matrix} f^*E & \rightarrow^{p_2} & E\\ \downarrow^{p_1} && \pi\\ X & \rightarrow^f &Y \end{matrix}\] By the universal property of a colimit, \(\pi\) will factor through morphisms \(E_j \to^{\pi_j} Y\), which will each fit into a pullback \(f^*E_j\), the direct sum of which will be the colimit.
In general \(f^*\) will not preserve limits, however it will preserve finite limits. The failure for infinite spaces is due to topological issues with taking arbitrary products.
We want to extend this structure of direct and inverse images to other spacial contexts.
Let \(X,Y\) be topoi. A geometric morphism from \(X\) to \(Y\) is a pair of adjoint functors \(f^*,f_*\) between them, such that \(f^*\) preserves finite limits. These are called the inverse resp. direct image of the geometric morphism.
We also want to distinguish those geometric morphisms where \(f^*\) itself admits a left adjoint \(f_!\). These are called essential geometric morphisms. The functor \(f_!\) is called the direct image with compact support. In a sense, it uses compactness to fix the failure of \(f^*\) to be compatible with arbitrary products.
We extend these definitions in the straightforward way to the case of geometric \(\infty\)-morphisms on \(\infty\)-topoi.
The payoff of all this machinery is the following: for some scheme \(X\), the topos of quasi-coherent sheaves on \(X\) admits the usual inverse and direct image functors \(f^*,f_*\), but they do not form an essentially geometric morphism. However, when we move to the setting of the \(\infty\)-topos of ind-coherent sheaves on a derived scheme \(X\), then \(f^*\) admits a left-adjoint \(f_!\). The existance and construction of this functor is highly nontrivial; a significant portion of the Gaitsgory book is devoted to pursuing it.
Why do we want such a left-adjoint \(f_!\) anyway?
One reason, coming from representation theory, is the following:
Let \(H \to^\phi G\) be groups. We can consider representations on them geometrically, as quasi-coherent sheaves on their classifying stacks \(BH,BG\), with corresponding functor \(\phi\).
Frobenius reciprocity tells us that induction \(\operatorname{ind}_{H \to G}^\phi\) of \(H\)-representations is left-adjoint to restriction \(\operatorname{res}_{G \to H}^\phi\) of \(G\)-representations.
In the geometric setting, restriction is given by the functor \(\phi^*: \operatorname{QCoh}(BG) \to \operatorname{QCoh}(BH)\). In order to recover the geometric induction in the general case, we need a left adjoint to \(\phi^*\): the functor \(\phi_!\) from before.
(This is a bit dishonest. In fact for nice compact Lie groups, it can be shown that \(f_!\) does indeed exist for the \(\operatorname{QCoh}\) categories. However it does not for less nice situations which arise in nature, such as exponentials of Lie algebras.)