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The six operations in topology

Marco Volpe

TL;DR

This work generalizes the six functor formalism for sheaves on locally compact Hausdorff spaces to coefficients in any stable bicomplete symmetric monoidal ∞-category, removing presentability/hypercompleteness constraints. It develops a robust framework built from Lurie’s tensor product of cocomplete ∞-categories, shape theory, and covariant Verdier duality to construct and control the five fundamental functors f_*, f^*, f_!, f^!, and their interrelations, including base-change and Künneth/projection formulas. A core technical achievement is establishing pullbacks with non-presentable coefficients via dualizability of spectral sheaves and Verdier duality, enabling a full six-functor formalism in this general setting. The paper also ties these constructions to relative Atiyah duality by expressing Thom spectra Th(E) through six-functor operations and showing f^!S_Y ≃ Th(T_f) for submersions, thus extending classical dualities to a broad ∞-categorical context with potential applications to generalized cohomology theories and motivic-style frameworks.

Abstract

In this paper we show that the six functor formalism for sheaves on locally compact Hausdorff topological spaces, as developed for example in Kashiwara and Schapira's book Sheaves on Manifolds, can be extended to sheaves with values in any closed symmetric monoidal $\infty$-category which is stable and bicomplete. Notice that, since we do not assume that our coefficients are presentable or restrict to hypercomplete sheaves, our arguments are not obvious and are substantially different from the ones explained by Kashiwara and Schapira. Along the way we also study locally contractible geometric morphisms and prove that, if $f:X\rightarrow Y$ is a continuous map which induces a locally contractible geometric morphism, then the exceptional pullback functor $f^!$ preserves colimits and can be related to the pullback $f^*$. At the end of our paper we also show how one can express Atiyah duality by means of the six functor formalism.

The six operations in topology

TL;DR

This work generalizes the six functor formalism for sheaves on locally compact Hausdorff spaces to coefficients in any stable bicomplete symmetric monoidal ∞-category, removing presentability/hypercompleteness constraints. It develops a robust framework built from Lurie’s tensor product of cocomplete ∞-categories, shape theory, and covariant Verdier duality to construct and control the five fundamental functors f_*, f^*, f_!, f^!, and their interrelations, including base-change and Künneth/projection formulas. A core technical achievement is establishing pullbacks with non-presentable coefficients via dualizability of spectral sheaves and Verdier duality, enabling a full six-functor formalism in this general setting. The paper also ties these constructions to relative Atiyah duality by expressing Thom spectra Th(E) through six-functor operations and showing f^!S_Y ≃ Th(T_f) for submersions, thus extending classical dualities to a broad ∞-categorical context with potential applications to generalized cohomology theories and motivic-style frameworks.

Abstract

In this paper we show that the six functor formalism for sheaves on locally compact Hausdorff topological spaces, as developed for example in Kashiwara and Schapira's book Sheaves on Manifolds, can be extended to sheaves with values in any closed symmetric monoidal -category which is stable and bicomplete. Notice that, since we do not assume that our coefficients are presentable or restrict to hypercomplete sheaves, our arguments are not obvious and are substantially different from the ones explained by Kashiwara and Schapira. Along the way we also study locally contractible geometric morphisms and prove that, if is a continuous map which induces a locally contractible geometric morphism, then the exceptional pullback functor preserves colimits and can be related to the pullback . At the end of our paper we also show how one can express Atiyah duality by means of the six functor formalism.

Paper Structure

This paper contains 21 sections, 62 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Linear overview
  3. Acknowledgements
  4. Sheaves and tensor products
  5. Tensor product of cocomplete $\infty$-categories
  6. Sheaves and cosheaves
  7. Shape theory and shape submersions
  8. Relative shape
  9. Locally contractible geometric morphisms
  10. Shape submersions
  11. Localization sequences
  12. Pullbacks with stable bicomplete coefficients
  13. Recollections on covariant Verdier duality
  14. Dualizability of spectral sheaves
  15. The pullback $f^{\ast}_{\mathscr{C}}$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\mathscr{C}$ be a stable bicomplete $\infty$-category, and let $X$ be a locally compact Hausdorff space. Then there is an equivalence where $\otimes$ on the left-hand side denotes the tensor product of stable cocomplete $\infty$-categories.

Theorems & Definitions (168)

  • Theorem 1.1: \ref{['shsptensCtoshC']}
  • Theorem 1.2: \ref{['properpsfcocont']}, \ref{['proppshtens']}
  • Theorem 1.3: \ref{['monpb']}, \ref{['proj']}, \ref{['kunneth']}
  • Remark 1.4
  • Theorem 1.5: \ref{['smoothpb']}
  • Theorem 1.6: \ref{['relatiyahdual']}
  • Remark 2.2
  • Remark 2.3
  • Remark 2.5
  • Remark 2.6
  • ...and 158 more