Condensed Mathematics: Pyknotic & Applications

Job: give the pyknotic approach of Barwick–Haine as an accessible / small-equivalent presentation of condensed sets via compact Hausdorff spaces, state the comparison theorem pyknotic \(\Leftrightarrow\) condensed, and survey the applications: condensed \(p\)-adic groups, condensed algebraic geometry over analytic rings, condensed Riemann–Hilbert / six-functor formalism, and the connections to prismatic cohomology and THH/TC.

Core

Pyknotic sets

Barwick–Haine's pyknotic mathematics replaces the site \(\mathrm{Prof}_{\kappa}\) of \(\kappa\)-small profinite sets by the site \(\mathrm{CHaus}_{\kappa}\) of \(\kappa\)-small compact Hausdorff spaces with the extensive topology (covers are finite families \(\{K_i\to K\}\) with \(K_i\to K\) closed inclusions inducing a disjoint-union effective epi). A pyknotic set is a sheaf

\[ X:\mathrm{CHaus}_{\kappa}^{\mathrm{op}}\to\mathrm{Set},\qquad K\mapsto X(K). \]

The category \(\mathrm{Pyk}(\mathrm{Set}):=\mathbf{Sh}(\mathrm{CHaus}_{\kappa},\text{extensive})\) is again a Grothendieck topos. The motivation is set-theoretic convenience: \(\mathrm{CHaus}\) is an accessible category with a small dense subcategory (the Stone spaces, equivalently profinite sets), so the topos \(\mathrm{Pyk}(\mathrm{Set})\) can be presented without choosing a Grothendieck universe — it is "the accessible version of condensed sets."

Comparison pyknotic \(\leftrightarrow\) condensed

The inclusion \(\mathrm{Prof}_{\kappa}\hookrightarrow\mathrm{CHaus}_{\kappa}\) (profinite \(\subset\) compact Hausdorff) induces by restriction a functor

\[ (-)|_{\mathrm{Prof}}:\mathrm{Pyk}(\mathrm{Set})\to\mathrm{Cond}(\mathrm{Set}),\qquad X\mapsto X|_{\mathrm{Prof}}. \]

Conversely, profinite sets are a dense subcategory of \(\mathrm{CHaus}\) (every compact Hausdorff is an inverse limit of profinite approximations / a filtered limit of finite CW approximations), so a sheaf on profinite sets extends canonically to one on compact Hausdorff.

Theorem (Barwick–Haine, Pyknotic) Statement
Restriction is an equivalence (modulo size) $(-)
Pyknotic abelian groups \(\mathrm{Pyk}(\mathrm{Ab}):=\mathrm{Ab}(\mathrm{Pyk}(\mathrm{Set}))\) is abelian, enough projectives, AB5-failure on non-ED witness — identical homological algebra to \(\mathrm{Cond}(\mathrm{Ab})\)
Solid / liquid transfer the analytic rings \(\overline{\mathbb{Z}},\mathbb{R}^{\triangleright},\dots\) and their module categories are defined identically in pyknotic; the liquid tensor product theorem holds verbatim
Compact Hausdorff projectives \(\mathbb{Z}[K]\) for \(K\) extremally disconnected compact Hausdorff are projective generators — same ED restriction as in condensed

The equivalence is not an identity on the nose: a pyknotic set carries the additional data of its values on non-profinite compact Hausdorff spaces, which are determined by the profinite restriction via sheafification but not specified point-by-point. For all homological-algebraic purposes the two presentations are interchangeable; the pyknotic variant is preferred when one wants to avoid universe bookkeeping or to use compact-Hausdorff-specific constructions (e.g. the Stone duality / Boolean-algebra dictionary, the Gelfand–Naimark duality).

Applications

1. Condensed \(p\)-adic groups. For a \(p\)-adic Lie group \(G\) (e.g. \(G=\mathrm{GL}_n(\mathbb{Z}_p)\), \(\mathrm{Gal}(\overline{K}/K)\)), the category of continuous \(G\)-representations on condensed abelian groups is a Grothendieck abelian category, and one can do group cohomology \(R\Gamma(G,M)\) in \(D(\mathrm{Cond}(\mathrm{Ab}))\) (or in \(D(\overline{\mathbb{Z}})\), \(D(\mathbb{Z}_p^{\blacksquare})\) after solid completion). This recovers continuous group cohomology of \(p\)-adic groups without the pathologies of the category of topological \(G\)-modules; the comparison with classical continuous cohomology is via the solid completion. The frontier is the condensed version of the \(p\)-adic local Langlands correspondence, where representations live naturally as solid \(\mathbb{Z}_p\)-modules rather than topological vector spaces.

2. Condensed algebraic geometry. Replacing discrete rings by condensed rings (rings in \(\mathrm{Cond}(\mathrm{Set})\)) and then by analytic rings (\(\overline{\mathbb{Z}}\), \(A_{\inf}^{\blacksquare}\), \(\mathbb{R}^{\triangleright}\), …) gives a condensed / analytic version of algebraic geometry in which the structure sheaf is an analytic ring and quasi-coherent sheaves are solid/liquid modules. This is the framework in which one expects to do over-analytic geometry: schemes over \(\overline{\mathbb{Z}}\) (the "solid" geometric world), over \(\mathbb{R}^{\triangleright}\) (the "liquid" archimedean world), over \(A_{\inf}^{\blacksquare}\) (the prismatic-perfectoid world). The six-functor formalism is being developed in this setting; the key point is that the solid/liquid tensor product is exact, so the usual Grothendieck–Verdier duality package applies without the AB5 obstruction.

3. Condensed Riemann–Hilbert / six-functor formalism. For a locally compact Hausdorff space \(X\) (e.g. a real manifold, a \(p\)-adic analytic space), the category of condensed sheaves on \(X\) and the condensed derived category \(D(\mathrm{Cond}(X;\mathbb{Z}))\) support the six functors \(f^*,f_*,\otimes,\mathrm{RHom},\otimes^!,f_!\) with the expected formalism — because the condensed category is a topos with enough points and the solid/liquid refinements are exact localizations. The condensed Riemann–Hilbert correspondence identifies flat connections (in the appropriate condensed sense) with locally constant condensed sheaves; the \(p\)-adic version (over \(\mathbb{Q}_p^{\triangleright}\)) is the active research program linking condensed mathematics to the geometric Langlands program on the \(p\)-adic side.

4. Connection to prismatic cohomology. The BMS prism \(A_{\inf}=W(R^\flat)\) (see prismatic foundations and analytic rings) is, in the condensed perspective, the analytic ring \(A_{\inf}^{\blacksquare}\); the BMS prismatic complex \(\triangle_{X/A_{\inf}}\) is a complex of solid \(A_{\inf}\)-modules, and the Frobenius structure is an analytic-ring endomorphism. The condensed viewpoint clarifies why the BMS construction works: \(A_{\inf}\) is solid-complete (a complete topological ring, hence solid), and the derived solid category over \(A_{\inf}^{\blacksquare}\) is AB5-exact, allowing the comparison isomorphisms (base changes along \(A_{\inf}^{\blacksquare}\to\mathcal{O}_K^{\blacksquare}\), \(\to B_{\mathrm{dR}}^{\triangleright}\), etc.) to be exact. This is the reformulation of prismatic cohomology as a solid cohomology theory.

5. Connection to THH / TC. Topological Hochschild homology and topological cyclic homology (Bökstedt–Hsiang–Madsen, Nikolaus–Scholze) are naturally condensed / pyknotic spectra: THH of a ring \(R\) is a cyclotomic spectrum, and the condensed viewpoint packages the Frobenius / Verschiebung / Tate operations as structure on a condensed object. The Nikolaus–Scholze formula for TC in terms of equalizers of Frobenius is naturally a statement in the condensed \(\infty\)-category, and the trace methods connecting TC to \(p\)-adic \(K\)-theory (see K-theory) are most cleanly stated in the condensed/solid language — the cyclotomic trace lands in solid \(A_{\inf}\)-modules, matching the BMS side.

See it

Pyknotic \(\Leftrightarrow\) condensed equivalence on \(S^1\) (a non-trivial compact Hausdorff space). Let \(K=S^1\) (the circle, a compact Hausdorff space, not profinite). A pyknotic set \(X\) has a value \(X(S^1)\); under the restriction-to-profinite equivalence, \(X\) is determined by \(\{X(S)\mid S\in\mathrm{Prof}\}\). To recover \(X(S^1)\) from the profinite values, use that \(S^1=\varprojlim_n S^1_n\) where \(S^1_n=S^1/(\text{subgroup of }n\text{-th roots})\) are finite (the finite cyclic covers / quotients); then by the sheaf condition on \(\mathrm{CHaus}\),

\[ X(S^1)=\varprojlim_n X(S^1_n), \]

an inverse limit over the profinite tower of finite quotients of \(S^1\). Concretely for \(X=\mathbb{Z}[-]\) the free condensed abelian group: \(\mathbb{Z}[S^1]\) as a pyknotic abelian group is \(\varprojlim_n\mathbb{Z}[S^1_n]=\varprojlim_n C(S^1_n,\mathbb{Z})\), the group of integral-valued functions on \(S^1\) that factor through some finite quotient — i.e. the locally constant integer-valued functions on \(S^1\), equivalently \(H^0_{\mathrm{sing}}(S^1;\mathbb{Z})\) in the condensed sense. This is the pyknotic counterpart of the condensed computation \(\mathbb{Z}[\mathbb{Z}_p]^{\blacksquare}\cong C(\mathbb{Z}_p,\mathbb{Z}_p)\) (see Solid Modules), and the equivalence pyknotic \(\Leftrightarrow\) condensed is the statement that these two computations (over \(\mathrm{CHaus}\) and over \(\mathrm{Prof}\)) give the same abelian group via the dense-profinite-extension.

Check

The pyknotic–condensed equivalence is a localization, not an identity, in the universe-agnostic presentation. Pin down exactly what extra data a pyknotic set carries that a condensed set does not, and why this data is determined by the profinite restriction.

A pyknotic set \(X\) specifies \(X(K)\) for every compact Hausdorff \(K\), including non-profinite \(K\) such as \(S^1\), \([0,1]\), \(\mathrm{GL}_n(\mathbb{Z}_p)\). A condensed set specifies only \(X(S)\) for profinite \(S\). The equivalence says: given the profinite values satisfying the sheaf condition on \(\mathrm{Prof}\), there is a unique extension to a sheaf on \(\mathrm{CHaus}\), given by \(X(K)=\varprojlim_{S\to K}X(S)\) over profinite approximations \(S\to K\) (profinite sets are dense in \(\mathrm{CHaus}\) under the extensive topology). The "extra data" \(X(K)\) is therefore not independent — it is forced by the profinite values via the sheaf condition. The localization (rather than identity) arises because, without fixing universes, the functor \((-)|_{\mathrm{Prof}}\) is essentially surjective but not literally bijective on objects: a condensed set and its canonical pyknotic extension are identified by the equivalence, but a naive (non-sheaf) presheaf on \(\mathrm{CHaus}\) extending a condensed set would not be a pyknotic set. The clean statement: the categories are equivalent; a pyknotic set is a condensed set plus the canonical extension, and the canonical extension is unique.

Why does the solid (resp. liquid) tensor product theorem hold verbatim in pyknotic, given that the projective generators \(\mathbb{Z}[S]^{\blacksquare}\) are indexed by profinite / ED sets in both cases?

The solid completion \(\mathbb{Z}[S]^{\blacksquare}=\varprojlim_n\mathbb{Z}[S_n]\) depends only on the profinite presentation of \(S\); for a compact Hausdorff \(K\), the corresponding pyknotic solid module is \(\mathbb{Z}[K]^{\blacksquare}=\varprojlim_{S\to K}\mathbb{Z}[S]^{\blacksquare}\) over profinite approximations, and the ED-projective-generator hypothesis is identical in both sites (extremally disconnected compact Hausdorff spaces = ED profinite sets by Stone duality). Hence the projective resolutions, the localization, and the exactness of the solid tensor product are the same computation in the two equivalent categories — the liquid tensor product theorem (the hard archimedean estimate) is invariant under the equivalence because it is a statement about the localized stable \(\infty\)-category, which is equivalence-invariant. One proves the theorem once (in either presentation) and transfers it.

Pitfalls

  • Pyknotic vs condensed equivalence is a localization, not an identity: a pyknotic set is not literally a condensed set with extra freely-chosen data on non-profinite \(K\); the values on non-profinite \(K\) are determined by sheafification. Treating \(\mathrm{Pyk}(\mathrm{Set})\) and \(\mathrm{Cond}(\mathrm{Set})\) as different categories with different objects (rather than equivalent presentations) is the standard confusion.
  • Set-theoretic universes: condensed sets require fixing a Grothendieck universe \(\kappa\); pyknotic sets are designed to be accessible (universe-agnostic) but one must still fix a size bound for the site. Conflating "accessible" with "no size choice at all" produces size errors. The safe statement: pyknotic avoids universe bookkeeping in the homological algebra but does not eliminate the need for a size bound on the site.
  • Solid vs liquid — which analytic ring (still): the pyknotic/condensed equivalence does not change the analytic-ring choice. A pyknotic \(\mathbb{R}\)-module is solid over \(\overline{\mathbb{Z}}\) or liquid over \(\mathbb{R}^{\triangleright}\) — these are genuinely different, and the choice is dictated by whether \(p\) is inverted / the problem is archimedean (see Analytic Rings). The equivalence of sites does not collapse the analytic-ring choice.
  • Non-archimedean vs archimedean liquid: \(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\) and \(\mathbb{Q}_p^{\triangleright}\) are \(p\)-invertible liquid rings (non-archimedean in flavor but liquid because \(p\) is inverted); \(\mathbb{R}^{\triangleright}\) is the archimedean liquid ring. The liquid tensor product theorem is proved in the archimedean case and deduced for the \(p\)-invertible ones — do not assume the \(p\)-invertible cases are easier; they depend on the archimedean theorem.
  • Condensed AG is research, not a finished theory: condensed / over-analytic algebraic geometry is an active program; the six-functor formalism in the condensed setting is not yet fully established in print. Use it as a pointer / framework, not as a library of proven theorems.

Refs

  • Barwick–Haine, Pyknotic Mathematics (preprint), §1–3 (pyknotic sets; the equivalence)
  • Clausen–Scholze, Condensed Mathematics / Analytic (the source for the solid/liquid theorems transferred to pyknotic)
  • Bhatt–Morrow–Scholze, Integral \(p\)-adic Hodge Theory §9 (\(A_{\inf}\) comparison; condensed perspective)
  • Nikolaus–Scholze, On topological cyclic homology (THH/TC as cyclotomic spectra — condensed natural habitat)

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Condensed index · Foundations · Solid Modules · Analytic Rings · Prismatic cohomology · K-theory · Topos theory · Functional analysis