Six Functor Formalisms

Table of Contents

• 1. Introduction
• 2. Setup: Sheaves on Locally Compact Spaces
  • 2.1 The Category of Spaces
  • 2.2 Abelian Sheaves
  • 2.3 The Derived Category D(X, Z)
• 3. The First Two Functors: Pullback and Pushforward
  • 3.1 Pullback f*
  • 3.2 Pushforward f_*
  • 3.3 The Adjunction f* ⊣ f_*
  • 3.4 Proper Base Change
• 4. The Tensor Product and Internal Hom
  • 4.1 Symmetric Monoidal Structure on D(X, Z)
  • 4.2 The Kunneth Formula
  • 4.3 The Projection Formula for Proper Maps
• 5. The Exceptional Functors: f_! and f^!
  • 5.1 Why f_* Is Not Enough
  • 5.2 Proper Pushforward f_!
  • 5.3 Compactly Supported Cohomology and the Open-Closed Triangle
  • 5.4 Exceptional Pullback f^!
  • 5.5 Verdier Duality and the Dualizing Functor
• 6. What Is a Six-Functor Formalism
  • 6.1 The Abstract Setup
  • 6.2 The Key Compatibilities
  • 6.3 The Problem of Higher Coherences
• 7. The Betti Space Construction
• 8. References

1. Introduction

📐 The central question these notes address is deceptively simple: what is a 6-functor formalism? For decades, the answer was informal — “you know it when you see it.” The first instance appeared in the development of étale cohomology by Grothendieck and his collaborators in the 1960s, and it was quickly recognized that analogous structures existed for sheaves on topological spaces, for D-modules on algebraic varieties in characteristic zero, and in many other geometric settings. Despite the ubiquity of these examples, a precise axiomatization capturing all the implicit structure proved elusive for many years.

This note builds up the theory from scratch through the lens of classical sheaf theory on locally compact Hausdorff spaces. This is the oldest and most geometrically intuitive setting, and it exhibits all six functors with minimal technical overhead. The goal is to:

1. Introduce each of the six functors in turn, motivating why each one is forced upon us by the geometry.
2. State the key theorems relating them: proper base change, the Künneth formula, the projection formula, and Verdier duality.
3. Arrive at a rough definition of what an abstract 6-functor formalism is, and explain why making this definition precise requires ∞-categorical machinery.

2. Setup: Sheaves on Locally Compact Spaces

2.1 The Category of Spaces

We work with the category \(\mathcal{C}\) of finite-dimensional locally compact Hausdorff spaces, specifically those that can be realized as locally closed subsets of some \(\mathbb{R}^n\). This includes: - Open and closed subsets of \(\mathbb{R}^n\) (e.g., balls, spheres) - Compact manifolds - Algebraic varieties over \(\mathbb{R}\) or \(\mathbb{C}\) (via their underlying topological spaces) - Totally disconnected spaces like the Cantor set \(2^{\mathbb{N}}\)

The Hausdorff condition ensures that diagonals \(X \hookrightarrow X \times X\) are closed embeddings, and finite-dimensionality is a technical hypothesis for the existence of \(f^!\).

2.2 Abelian Sheaves

For \(X \in \mathcal{C}\), we write \(\mathrm{Ab}(X)\) for the abelian category of abelian sheaves on \(X\). Recall that a sheaf of abelian groups \(\mathcal{F}\) on \(X\) assigns an abelian group \(\mathcal{F}(U)\) to each open \(U \subseteq X\), together with restriction maps, subject to the sheaf condition: for any open cover \(\{U_i\}\) of \(U\),

\[0 \to \mathcal{F}(U) \to \prod_i \mathcal{F}(U_i) \rightrightarrows \prod_{i,j} \mathcal{F}(U_i \cap U_j)\]

is exact. The stalk of \(\mathcal{F}\) at a point \(x \in X\) is \(\mathcal{F}_x = \varinjlim_{U \ni x} \mathcal{F}(U)\).

Example. The constant sheaf \(\underline{M}\) associated to an abelian group \(M\) assigns locally constant \(M\)-valued functions to each open \(U\).

2.3 The Derived Category D(X, Z)

The derived category \(D(X, \mathbb{Z}) := D(\mathrm{Ab}(X))\) is the localization of the category of cochain complexes of abelian sheaves at quasi-isomorphisms. Morphisms in \(D(X, \mathbb{Z})\) are roofs — formal compositions of chain maps with quasi-isomorphisms inverted — so two complexes that are quasi-isomorphic become genuinely isomorphic.

The key additional structure is the triangulated structure: for any map \(f : A \to B\) in \(D(X, \mathbb{Z})\), there is a distinguished triangle

\[A \to B \to C \xrightarrow{+1}\]

where \(C = \mathrm{Cone}(f)\) is the mapping cone. Distinguished triangles play the role of short exact sequences; they give long exact sequences in cohomology.

As a graded invariant, the cohomology \(H^i(X, \mathbb{Z})\) is extracted by applying the global sections functor

\[\Gamma(X, -) = H^0(X, -) : \mathrm{Ab}(X) \to \mathrm{Ab},\]

which is left exact, and taking its right derived functors \(R^i\Gamma(X, -) = H^i(X, -)\). The correct derived version packages this as a single functor

\[R\Gamma(X, -) : D(X, \mathbb{Z}) \to D(\mathbb{Z}).\]

Applied to the constant sheaf \(\underline{\mathbb{Z}}\), this recovers the sheaf cohomology groups \(H^i(X, \mathbb{Z})\).

3. The First Two Functors: Pullback and Pushforward

3.1 Pullback f*

For a continuous map \(f : X \to Y\), there is an exact functor

\[f^* : \mathrm{Ab}(Y) \to \mathrm{Ab}(X)\]

defined by \((f^*\mathcal{F})_x = \mathcal{F}_{f(x)}\) on stalks (or more precisely, by sheafifying the presheaf \(U \mapsto \varinjlim_{V \supseteq f(U)} \mathcal{F}(V)\)).

Since \(f^*\) is exact, it induces a functor at the derived level:

\[f^* : D(Y, \mathbb{Z}) \to D(X, \mathbb{Z}) \qquad \textbf{(Functor 1)}\]

Example. If \(f : X \to \{*\}\) is the terminal map, then \(f^*\mathbb{Z} = \underline{\mathbb{Z}}\) is the constant sheaf on \(X\).

Example. If \(i : Z \hookrightarrow X\) is a closed inclusion, then \(i^*\mathcal{F} = \mathcal{F}|_Z\) is just the restriction of \(\mathcal{F}\) to \(Z\).

3.2 Pushforward f_*

The pushforward is the right adjoint to pullback. For \(f : X \to Y\) and \(\mathcal{F} \in \mathrm{Ab}(X)\), define

\[(f_*\mathcal{F})(V) = \mathcal{F}(f^{-1}(V)) \quad \text{for open } V \subseteq Y.\]

This is already a sheaf (no sheafification needed), and the assignment \(\mathcal{F} \mapsto f_*\mathcal{F}\) is a left exact functor. Its right derived functor is

\[Rf_* : D(X, \mathbb{Z}) \to D(Y, \mathbb{Z}) \qquad \textbf{(Functor 2)}\]

Example. If \(f : X \to \{*\}\) is the terminal map, then \(f_* = R\Gamma(X, -)\) is the global sections functor. The cohomology of \(X\) is \(H^i(X, \mathbb{Z}) = H^i(Rf_*\underline{\mathbb{Z}})\).

Example. For \(f : \mathbb{R} \to \{*\}\) and \(\mathcal{F} = \underline{\mathbb{Z}}\), we get \(Rf_*\underline{\mathbb{Z}} = R\Gamma(\mathbb{R}, \mathbb{Z})\). Since \(\mathbb{R}\) is contractible, this is just \(\mathbb{Z}\) concentrated in degree 0.

3.3 The Adjunction f* ⊣ f_*

The fundamental adjunction is

\[\mathrm{Hom}_{D(X)}(f^*A, B) \cong \mathrm{Hom}_{D(Y)}(A, Rf_*B)\]

naturally in \(A \in D(Y, \mathbb{Z})\) and \(B \in D(X, \mathbb{Z})\).

The unit of this adjunction is a natural map \(\eta : A \to Rf_*f^*A\), and the counit is \(\varepsilon : f^*Rf_*B \to B\).

Key consequence. Taking \(f : X \to \{*\}\) and \(A = \mathbb{Z}\), the unit gives a map \(\mathbb{Z} \to R\Gamma(X, \underline{\mathbb{Z}})\), which is exactly the augmentation map encoding the cohomology of \(X\).

3.4 Proper Base Change

The pushforward \(Rf_*\) computes “global cohomology of the fibers” only when \(f\) is proper — meaning the preimage of any compact set is compact. The precise statement is:

Theorem 3.1 (Proper Base Change). Let \(f : X \to Y\) be a proper map in \(\mathcal{C}\) and let

be a Cartesian square (i.e., \(X' = X \times_Y Y'\)). Then for any \(A \in D(X, \mathbb{Z})\), the natural base change map

\[g^* Rf_* A \xrightarrow{\sim} Rf'_* g'^* A\]

is an isomorphism.

Stalk form. For \(g : \{y\} \hookrightarrow Y\) the inclusion of a point, proper base change says

\[(Rf_*A)_y \xrightarrow{\sim} R\Gamma(X_y, A|_{X_y})\]

where \(X_y = X \times_Y \{y\}\) is the fiber over \(y\). So the stalks of \(Rf_*A\) are exactly the cohomology of the fibers. This is the reason \(f\) must be proper: for non-proper \(f\), the stalk \((Rf_*A)_y\) sees contributions from cohomology “at infinity” in the fibers.

4. The Tensor Product and Internal Hom

4.1 Symmetric Monoidal Structure on D(X, Z)

Beyond just cohomology of individual sheaves, we want to understand how cohomology interacts with products. This forces us to equip each \(D(X, \mathbb{Z})\) with a tensor product.

The abelian category \(\mathrm{Ab}(X)\) has a tensor product: for \(\mathcal{F}, \mathcal{G} \in \mathrm{Ab}(X)\), the tensor product \(\mathcal{F} \otimes \mathcal{G}\) is the sheafification of the presheaf \(U \mapsto \mathcal{F}(U) \otimes_\mathbb{Z} \mathcal{G}(U)\). Since this is right exact but not left exact, we derive it:

\[- \otimes - : D(X, \mathbb{Z}) \times D(X, \mathbb{Z}) \to D(X, \mathbb{Z}) \qquad \textbf{(Functor 3)}\]

This is the derived tensor product. The unit object is \(\underline{\mathbb{Z}}\) (the constant sheaf). Together with the natural associativity, commutativity, and unit isomorphisms satisfying coherence axioms, this makes \(D(X, \mathbb{Z})\) a symmetric monoidal category.

The internal Hom is characterized by the adjunction

\[\mathrm{Hom}_{D(X)}(A \otimes B, C) \cong \mathrm{Hom}_{D(X)}(A, \mathcal{H}om(B, C))\]

and gives a functor

\[\mathcal{H}om(-, -) : D(X, \mathbb{Z})^{\mathrm{op}} \times D(X, \mathbb{Z}) \to D(X, \mathbb{Z}) \qquad \textbf{(Functor 4)}\]

Compatibility with pullback. The pullback \(f^*\) is symmetric monoidal: there is a natural isomorphism

\[f^*(A \otimes B) \xrightarrow{\sim} f^*A \otimes f^*B\]

compatible with all the monoidal structure. This is because \(f^*\) is exact and commutes with sheafification, so it distributes over the tensor product construction.

4.2 The Kunneth Formula

💡 The first payoff of having the tensor product is the Künneth formula.

Theorem 4.2 (Künneth Formula). For proper \(X, Y \in \mathcal{C}\) and any \(A \in D(X, \mathbb{Z})\), \(B \in D(Y, \mathbb{Z})\), there is a natural isomorphism

\[R\Gamma(X, A) \otimes R\Gamma(Y, B) \xrightarrow{\sim} R\Gamma(X \times Y, A \boxtimes B)\]

where \(A \boxtimes B := p_1^* A \otimes p_2^* B \in D(X \times Y, \mathbb{Z})\) is the external tensor product.

In the special case \(A = \underline{\mathbb{Z}}\), \(B = \underline{\mathbb{Z}}\), this gives \(H^*(X, \mathbb{Z}) \otimes H^*(Y, \mathbb{Z}) \cong H^*(X \times Y, \mathbb{Z})\) (as graded groups).

4.3 The Projection Formula for Proper Maps

The projection formula is a fundamental compatibility between the tensor product and pushforward:

Theorem 4.3 (Projection Formula). Let \(f : X \to Y\) be proper and let \(A \in D(X, \mathbb{Z})\), \(B \in D(Y, \mathbb{Z})\). There is a natural isomorphism

\[Rf_*A \otimes B \xrightarrow{\sim} Rf_*(A \otimes f^*B).\]

5. The Exceptional Functors: f_! and f^!

5.1 Why f_* Is Not Enough

🔑 The four functors \(f^*, Rf_*, \otimes, \mathcal{H}om\) are already rich enough to define cohomology and prove the Künneth formula. So why do we need two more?

The issue is duality. Consider \(f : X \to \{*\}\) for a compact oriented \(d\)-manifold \(X\). Poincaré duality says:

\[H^k(X, \mathbb{Z}) \cong H_{d-k}(X, \mathbb{Z}) \cong H^{d-k}(X, \mathbb{Z})^\vee\]

(the last isomorphism after tensoring with \(\mathbb{Q}\)). In sheaf-theoretic language, there should be a dualizing sheaf \(\omega_X \in D(X, \mathbb{Z})\) such that \(\mathcal{H}om(A, \omega_X)\) is the “Poincaré dual” of \(A\).

The problem is that \(\omega_X\) is not the constant sheaf in general — it detects local orientation. Moreover, to state a relative version of duality (for \(f : X \to Y\) rather than \(f : X \to \{*\}\)), one needs a functor \(f^!\) that is right adjoint to \(f_!\), and neither of these is \(Rf_*\) or \(f^*\).

A second issue: the base change and projection formula for \(Rf_*\) require properness. In many geometric applications, we need these compatibilities for all maps, not just proper ones. The functor \(f_!\) (defined below) satisfies base change and projection formula for all \(f\).

5.2 Proper Pushforward f_!

For \(f : X \to Y\) in \(\mathcal{C}\), the pushforward with proper support \(f_!\) is defined as follows. The sections with proper support of a sheaf \(\mathcal{F}\) over an open \(V \subseteq Y\) are

\[\Gamma_!(V, f, \mathcal{F}) = \{ s \in \Gamma(f^{-1}(V), \mathcal{F}) \mid \mathrm{supp}(s) \to V \text{ is proper}\}.\]

This defines a left exact functor \(f_! : \mathrm{Ab}(X) \to \mathrm{Ab}(Y)\), and we set

\[Rf_! : D(X, \mathbb{Z}) \to D(Y, \mathbb{Z}) \qquad \textbf{(Functor 5)}\]

to be its right derived functor.

Key properties: - There is a natural transformation \(Rf_! \to Rf_*\), which is an isomorphism when \(f\) is proper. - For an open embedding \(j : U \hookrightarrow X\), \(j_!\) is exact and extends sections by zero: \((j_!\mathcal{F})_x = 0\) for \(x \notin U\) and \(\mathcal{F}_x\) for \(x \in U\). - For a closed embedding \(i : Z \hookrightarrow X\), \(i_! = i_*\).

General base change and projection formula. Unlike \(Rf_*\), the functor \(Rf_!\) satisfies base change and projection formula without any properness assumption:

\[g^* Rf_! \xrightarrow{\sim} Rf'_! g'^* \qquad \text{and} \qquad Rf_!A \otimes B \xrightarrow{\sim} Rf_!(A \otimes f^*B)\]

hold for all \(f\) (not just proper). However, unlike the analogous results for \(Rf_*\), these are now extra data — not maps derived from adjunctions — and must be specified as part of the structure. This is the heart of the difficulty in formalizing a 6-functor formalism.

5.3 Compactly Supported Cohomology and the Open-Closed Triangle

🗜️ With \(f_!\) in hand, we can define one of the most important geometric invariants.

Definition. For \(p : X \to \{*\}\) the structure map, the cohomology with compact support of \(X\) is

\[H^k_c(X, \mathbb{Z}) := H^k(Rp_!\underline{\mathbb{Z}}).\]

For compact \(X\), \(p_! = p_*\) (since \(p\) is proper), so \(H^*_c(X) = H^*(X)\). The interesting case is non-compact \(X\).

Key examples:

\[H^k_c(\mathbb{R}^n, \mathbb{Z}) = \begin{cases} \mathbb{Z} & k = n \\ 0 & k \neq n \end{cases}\]

This is dramatically different from \(H^k(\mathbb{R}^n, \mathbb{Z}) = \mathbb{Z}\) concentrated in degree \(0\) (since \(\mathbb{R}^n\) is contractible). Intuitively, compactly supported cohomology “sees” the \(n\)-dimensional hole at infinity that ordinary cohomology cannot detect.

The open-closed exact triangles. Given \(j : U \hookrightarrow X\) an open embedding and \(i : Z = X \setminus U \hookrightarrow X\) the complementary closed embedding, the natural maps give two fundamental exact triangles in \(D(X, \mathbb{Z})\): for any \(\mathcal{F} \in D(X, \mathbb{Z})\),

\[j_!j^*\mathcal{F} \longrightarrow \mathcal{F} \longrightarrow i_*i^*\mathcal{F} \xrightarrow{+1} \qquad \textbf{(Triangle 1)}\]

\[i_*i^!\mathcal{F} \longrightarrow \mathcal{F} \longrightarrow Rj_*j^*\mathcal{F} \xrightarrow{+1} \qquad \textbf{(Triangle 2)}\]

Both triangles can be verified stalk-by-stalk:

So Triangle 1 is exact at every stalk: an isomorphism at \(U\)-points followed by zero, then zero followed by an isomorphism at \(Z\)-points.

Long exact sequence for a pair. Applying \(R\Gamma(X, -)\) to Triangle 1 with \(\mathcal{F} = \underline{\mathbb{Z}}_X\) and using that \(R\Gamma(X, j_!\underline{\mathbb{Z}}_U) = R\Gamma_c(U, \mathbb{Z})\) (when \(X\) is compact) gives the long exact sequence of the pair \((X, Z)\):

\[\cdots \to H^k_c(U, \mathbb{Z}) \to H^k(X, \mathbb{Z}) \to H^k(Z, \mathbb{Z}) \to H^{k+1}_c(U, \mathbb{Z}) \to \cdots\]

5.4 Exceptional Pullback f^!

The functor \(Rf_!\) has a right adjoint:

\[f^! : D(Y, \mathbb{Z}) \to D(X, \mathbb{Z}) \qquad \textbf{(Functor 6)}\]

called the exceptional inverse image or extraordinary pullback, characterized by

\[\mathrm{Hom}_{D(Y)}(Rf_!A, B) \cong \mathrm{Hom}_{D(X)}(A, f^!B).\]

Key computations: - For an open embedding \(j : U \hookrightarrow X\): since \(j_! \dashv j^*\), we have \(j^! = j^*\). - For a closed embedding \(i : Z \hookrightarrow X\): \(i^!\mathcal{F}\) computes local sections of \(\mathcal{F}\) supported on \(Z\) — the local cohomology of \(\mathcal{F}\) with support in \(Z\). - For a proper map \(f\): since \(Rf_! \cong Rf_*\), the adjunction \(Rf_! \dashv f^!\) gives \(f^!\) as a right adjoint of \(Rf_*\).

5.5 Verdier Duality and the Dualizing Functor

With all six functors in place, we can state and prove duality.

Definition. For a map \(f : X \to Y\), the relative dualizing complex is \(\omega_{X/Y} = f^!\underline{\mathbb{Z}}_Y \in D(X, \mathbb{Z})\). For \(p : X \to \{*\}\), write \(\omega_X = p^!\mathbb{Z}\) for the absolute dualizing complex of \(X\).

Theorem 5.4 (Verdier Duality, Local Form). Let \(f : X \to Y\) be a “manifold bundle” of relative dimension \(d\): locally on \(X\), the map looks like \(Y \times B^d \to Y\) (where \(B^d\) is an open \(d\)-ball). Then \(f^! \cong f^*(-) \otimes \omega_{X/Y}\) where \(\omega_{X/Y}\) is locally isomorphic to \(\underline{\mathbb{Z}}[d]\).

Corollary (Global Poincaré Duality). If \(X\) is a compact oriented \(d\)-manifold and \(f : X \to \{*\}\), then \(\omega_X \cong \underline{\mathbb{Z}}[d]\) globally, and for any \(A \in D(X, \mathbb{Z})\), \(B \in D(\mathbb{Z})\):

\[R\mathcal{H}om_{D(\mathbb{Z})}(R\Gamma(X, A), B) \cong R\Gamma(X, \mathcal{H}om(A, f^!B)).\]

The Verdier duality functor. The dualizing complex \(\omega_X = p^!\mathbb{Z}\) assembles into a contravariant involution:

Definition. The Verdier duality functor is

\[\mathbb{D}_X : D(X, \mathbb{Z})^{\mathrm{op}} \to D(X, \mathbb{Z}), \quad \mathbb{D}_X(A) = R\mathcal{H}om(A, \omega_X).\]

For an oriented \(d\)-manifold, \(\omega_X = \underline{\mathbb{Z}}[d]\), so \(\mathbb{D}_X(A) = R\mathcal{H}om(A, \underline{\mathbb{Z}})[d]\).

Theorem (Biduality). For any \(A\) in the bounded derived category \(D^b_c(X, \mathbb{Z})\) of constructible sheaves, the natural evaluation map

\[A \xrightarrow{\sim} \mathbb{D}_X(\mathbb{D}_X(A))\]

is an isomorphism. In other words, \(\mathbb{D}_X\) is an involution on \(D^b_c(X, \mathbb{Z})\).

Duality exchanges \(f_!\) and \(f^!\). The most important property of \(\mathbb{D}\) is that it interchanges the two exceptional functors. For any \(f : X \to Y\) in \(\mathcal{E}\), there are natural isomorphisms:

\[Rf_* \circ \mathbb{D}_X \cong \mathbb{D}_Y \circ Rf_! \qquad \text{and} \qquad f^! \circ \mathbb{D}_Y \cong \mathbb{D}_X \circ f^*.\]

Informally: computing \(Rf_*\) after dualizing on \(X\) is the same as first computing \(Rf_!\), then dualizing on \(Y\). This is the precise sense in which \(f_!\) and \(f^!\) are “adjoint” not just to each other, but dual to \(f^*\) and \(Rf_*\) via the dualizing functor.

Consequence: duality between ordinary and compact cohomology. Applying \(R\Gamma(Y, -)\) and using \(\mathbb{D}_Y \circ Rf_! \cong Rf_* \circ \mathbb{D}_X\), one obtains (for \(A \in D^b_c(X)\) with free cohomology sheaves):

\[H^k(X, A)^\vee \cong H^{-k}_c(X, \mathbb{D}_X(A)).\]

For a compact oriented \(d\)-manifold with \(A = \underline{\mathbb{Z}}\), \(\mathbb{D}_X(\underline{\mathbb{Z}}) = \underline{\mathbb{Z}}[d]\), so this gives \(H^k(X, \mathbb{Z})^\vee \cong H^{d-k}(X, \mathbb{Z})\) — classical Poincaré duality.

6. What Is a Six-Functor Formalism

6.1 The Abstract Setup

We can now give a rough definition of an abstract 6-functor formalism. The data consists of:

• A category \(\mathcal{C}\) of “geometric objects” (topological spaces, schemes, analytic spaces, stacks, …).
• A class \(\mathcal{E}\) of “admissible” morphisms in \(\mathcal{C}\) (those for which \(f_!\) and \(f^!\) exist). We require: \(\mathcal{E}\) is stable under composition, pullback, and contains all isomorphisms.
• An assignment \(X \mapsto D(X)\) from \(\mathcal{C}\) to stable \(\infty\)-categories (or at least triangulated categories).
• Six functors:

The adjunctions are:

graph LR
    A["f* (pullback)"] -->|"left adjoint of"| B["Rf* (pushforward)"]
    C["tensor A"] -->|"left adjoint of"| D["Hom(A, -)"]
    E["Rf! (shriek push)"] -->|"left adjoint of"| F["f! (shriek pull)"]

6.2 The Key Compatibilities

Beyond the adjunctions, the key structural constraints are:

1. Pullback is symmetric monoidal: Natural isomorphisms \(f^*(A \otimes B) \xrightarrow{\sim} f^*A \otimes f^*B\), compatible with composition and with the symmetric monoidal axioms.

2. Composition: For \(f : X \to Y\) and \(g : Y \to Z\) both in \(\mathcal{E}\), natural isomorphisms \[R(g \circ f)_! \xrightarrow{\sim} Rg_! \circ Rf_!\] satisfying associativity for triple composites.

3. Base change: For any Cartesian square with \(f \in \mathcal{E}\): \[g^* Rf_! \xrightarrow{\sim} Rf'_! g'^*\]

4. Projection formula: For \(f \in \mathcal{E}\) and any \(A \in D(X)\), \(B \in D(Y)\): \[Rf_!A \otimes B \xrightarrow{\sim} Rf_!(A \otimes f^*B)\]

6.3 The Problem of Higher Coherences

⚠️ The real difficulty in axiomatizing a 6-functor formalism is not the list of isomorphisms, but the coherences between them.

Consider just the composition isomorphism \(h_! \cong g_! f_!\) for \(h = g \circ f\). For a triple composite \(k = h \circ g \circ f\), there are two ways to build the isomorphism \(k_! \cong h_! g_! f_!\): either compose the isomorphisms for \((f, g)\) then \((gf, h)\), or for \((g, h)\) then \((f, hg)\). These must agree.

But when \(D(X)\) is an \(\infty\)-category rather than a triangulated category, an isomorphism is not just a morphism — it is a path in the space of morphisms. Two paths can be homotopic in many ways, and homotopies between homotopies must also cohere, ad infinitum.

How this is resolved. The modern approach (Mann [Man22b], building on Liu–Zheng [LZ12a]) packages all these coherences by encoding \(f_!\) not as a separate functor but as part of a single functor

\[D : \mathrm{Corr}(\mathcal{C}, \mathcal{E}) \to \mathrm{Cat}_\infty\]

from an \(\infty\)-category of correspondences (or spans) over \(\mathcal{C}\), where \(f^*\) lives on backward arrows and \(f_!\) lives on forward arrows. The coherences of base change are then built into the categorical structure of correspondences automatically.

7. The Betti Space Construction

🌐 We close with a beautiful example showing that the 6-functor formalism for locally compact Hausdorff spaces is secretly of “algebraic” origin.

For \(X \in \mathcal{C}\), define the functor

\[X_{\mathrm{Betti}} : \mathrm{Schemes}^{\mathrm{op}} \to \mathrm{Sets}, \quad S \mapsto \{\text{continuous maps } |S| \to X\},\]

where \(|S|\) denotes the underlying topological space of the scheme \(S\).

Theorem 7.1 (Scholze). \(X_{\mathrm{Betti}}\) is representable as a pro-étale algebraic space: it admits a pro-étale surjection from a scheme. Moreover, there is a natural equivalence

\[D(X, \mathbb{Z}) \xrightarrow{\sim} D_{\mathrm{qc}}(X_{\mathrm{Betti}})\]

where \(D_{\mathrm{qc}}\) denotes the derived category of quasi-coherent sheaves.

This example hints at a broader phenomenon: in many settings, there is a factorization

\[\mathcal{C} \xrightarrow{F} \{\text{analytic stacks}\} \xrightarrow{D_{\mathrm{qc}}} \mathrm{Cat}_\infty\]

where \(F\) takes each geometric object \(X\) to some kind of “ring stack” \(F(X)\), and the 6-functor formalism is pulled back from the formalism for quasi-coherent sheaves. This perspective, initiated by Drinfeld in the context of prismatic cohomology, provides a unified geometric picture of six-functor formalisms across topology, algebraic geometry, and \(p\)-adic analysis.

8. References