Condensed Mathematics¶
Job: re-architect topology, functional analysis, and homological algebra on the solid base of condensed sets — sheaves on the pro-étale site of a point (equivalently sheaves on profinite sets for the effective-epimorphism topology) — and on the analytic rings (Clausen–Scholze) that repair the failure of \(\mathrm{Cond}(\mathrm{Ab})\) to be exact, producing the solid and liquid categories where \(p\)-adic and archimedean functional analysis become algebra.
Core¶
Classical topology has an exactness defect: the category of topological abelian groups is not abelian (kernels/cokernels of continuous maps misbehave), and the category of compactly generated weak Hausdorff spaces — though convenient — does not interact cleanly with algebra. Condensed mathematics replaces "topological space" by a condensed set, i.e. a sheaf
on the site of profinite sets with the effective-epimorphism topology. The category \(\mathrm{Cond}(\mathrm{Set})\) is a Grothendieck topos (in a fixed universe); \(\mathrm{Cond}(\mathrm{Ab})\) is abelian with all colimits and enough projectives, but it is not AB4/AB5-exact on non-extremally-disconnected test sets — the measure-theoretic obstruction. The fix is to pass to the solid (\(\overline{\mathbb{Z}}\)) and liquid (\(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\), \(\mathbb{R}^{\triangleright}\)) subcategories, where the analytic-ring tensor product restores exactness.
| Layer | Content |
|---|---|
| Condensed sets | sheaves on profinite sets / pro-étale site of a point; \(\mathrm{Cond}(\mathrm{Set})\) a topos |
| Condensed abelian groups | \(\mathrm{Cond}(\mathrm{Ab})\) abelian, enough projectives; not AB5 on all profinite sets |
| Solid | analytic ring \(\overline{\mathbb{Z}}\); solid \(\mathbb{Z}\)-modules; solid tensor product \(\otimes_{\blacksquare}\) |
| Liquid | analytic rings \(\mathbb{R}^{\triangleright}\), \(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\); liquid tensor product theorem |
| Pyknotic | Barwick–Haine small-equivalent presentation; set-theoretic convenience |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | condensed sets as sheaves on profinite sets; \(\mathrm{Cond}(\mathrm{Set})\), \(\mathrm{Cond}(\mathrm{Ab})\); comparison with CGWH; the exactness obstruction |
| Solid Modules | solid abelian groups; \(\otimes_{\blacksquare}\); \(\mathbb{Z}[S]^{\blacksquare}\); the analytic ring \(\overline{\mathbb{Z}}\); dictionary to complete topological abelian groups |
| Analytic Rings | analytic rings \((\mathcal{A},\mathcal{A}[-])\); the standard table; liquid tensor product theorem; \(A_{\inf}\) and \(p\)-adic Hodge |
| Pyknotic & Applications | pyknotic sets (Barwick–Haine); pyknotic \(\Leftrightarrow\) condensed; condensed \(p\)-adic groups, condensed AG, Riemann–Hilbert, THH/TC |
Dependency order¶
- Topos theory: Grothendieck toposes, sheaves on sites, geometric morphisms — condensed sets are a sheaf topos; see topos foundations.
- Functional analysis: Banach/Hilbert/Fréchet, \(\ell^2\) — the dictionary to solid/liquid modules recovers these as algebra.
- Category theory: abelian categories, exactness axioms AB4/AB5, completions — the obstruction is a failure of AB5.
- Derived AG: animated rings, derived completion — analytic rings extend animated rings by a condensed structure sheaf.
- Prismatic cohomology: \(A_{\inf}=W(R^\flat)\) as an analytic ring; BMS in the condensed language.
- Foundations \(\to\) solid \(\to\) analytic rings \(\to\) pyknotic/applications.
Interfaces¶
| Area | Contact |
|---|---|
| Topos theory | \(\mathrm{Cond}(\mathrm{Set})=\mathbf{Sh}(\mathrm{Prof},\mathrm{eff.epi.})\) is a Grothendieck topos |
| Functional analysis | Banach/Hilbert/Fréchet spaces \(\leftrightarrow\) liquid \(\mathbb{R}\)-modules; \(\ell^2\) as solid/liquid object |
| Category theory | abelian category; AB4/AB5; enough projectives; stable \(\infty\)-categories |
| Derived AG | animated rings; analytic rings as condensed + animated; derived solid/liquid categories |
| Spectral AG | condensed \(\mathbb{E}_\infty\)-rings; analytic ring spectra |
| Prismatic cohomology | \(A_{\inf}^{\blacksquare}\) as analytic ring; BMS comparison in analytic-ring language |
| K-theory | condensed THH/TC; trace methods in the condensed setting |
Notation (paper-standard)¶
- \(\mathrm{Cond}(\mathrm{Set})\), \(\mathrm{Cond}(\mathrm{Ab})\) — condensed sets / abelian groups; \(\mathcal{C}=\mathrm{Cond}(\mathrm{Ab})\)
- \(S\) profinite; \(X(S)=\mathrm{Hom}(S,X)\) the \(S\)-points of a condensed set \(X\)
- \(\mathbb{Z}[S]\) free condensed abelian group on \(S\); \(\mathbb{Z}[S]^{\blacksquare}\) its solid completion
- \(\otimes_{\blacksquare}\) solid tensor product; \(\otimes_{\triangleright}\) liquid tensor product
- \(\overline{\mathbb{Z}}=(\mathbb{Z},\mathbb{Z}[-]^{\blacksquare})\) the solid analytic ring; \(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\), \(\mathbb{R}^{\triangleright}\), \(\mathbb{Q}_p^{\triangleright}\) liquid analytic rings
- \(A_{\inf}^{\blacksquare}\) the \(A_{\inf}\)-analytic ring (link to prismatic)
- \(\mathrm{Solid}_{\mathbb{Z}}\) the category of solid \(\mathbb{Z}\)-modules; \(\mathrm{Liq}_{\mathbb{R}}\) liquid \(\mathbb{R}\)-modules
See it¶
The exactness defect, concretely. In \(\mathrm{Cond}(\mathrm{Ab})\) the sequence \(0\to\mathbb{Z}\to\mathbb{R}\to\mathbb{R}/\mathbb{Z}\to 0\) (each given the condensed topology of the classical topological group) is exact, but tensoring with \(\mathbb{R}\) does not preserve exactness in degree \(1\) in the ordinary condensed tensor — \(\mathbb{R}\otimes_{\mathbb{Z}}\mathbb{R}\neq\mathbb{R}\) as a condensed abelian group, and \(\mathrm{Ext}^1_{\mathrm{Cond}}(\mathbb{R},\mathbb{Z})\) is enormous. This is the obstruction Clausen–Scholze trace to the non-exactness of \(\mathrm{Cond}(\mathrm{Ab})\) on non-extremally-disconnected sets; the solid (resp. liquid) tensor product is engineered to kill exactly this obstruction. See Solid Modules and Analytic Rings.
Check¶
Why is \(\mathrm{Cond}(\mathrm{Ab})\) abelian but not AB5-exact, and which test sets witness the failure?
\(\mathrm{Cond}(\mathrm{Ab})\) is the category of abelian group objects in the topos \(\mathrm{Cond}(\mathrm{Set})\), hence abelian with all colimits (topos = AB3) and enough projectives (profinite \(\mathbb{Z}[S]\) are projective generators). AB5 (filtered colimits exact) fails because filtered colimits of sheaves are sheaf-colimits, and sheafification is not exact on the eff-epi topology when the cover has non-extremally-disconnected nerves: the witness is the failure of \(\varprojlim^1\) to vanish on towers indexed by non-ED profinite sets, equivalently the non-vanishing of higher derived limits producing \(\mathrm{Ext}^1_{\mathrm{Cond}}(\mathbb{R},\mathbb{Z})\).
State the slogan that distinguishes solid from liquid in one line, in terms of which prime is invertible.
Solid = over \(\overline{\mathbb{Z}}\) (no prime inverted; the non-archimedean / \(p\)-adic side, complete topological abelian groups); liquid = over \(\mathbb{R}^{\triangleright}\) or \(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\) (the relevant prime invertible; the archimedean / \(p\)-invertible side, where the hard exactness theorem lives). Confusing the two is the standard error.
Pitfalls¶
- Condensed \(\neq\) topological: a condensed set is a sheaf, not a space; the comparison with CGWH is full faithful on a restricted class and the functor "underlying topological space" is not the inverse on all objects.
- Set-theoretic universe: \(\mathrm{Cond}(\mathrm{Set})\) must be taken in a fixed Grothendieck universe (or via the pyknotic accessible variant); ignoring this produces size errors in "the topos of all condensed sets."
- Solid vs liquid: which analytic ring — \(\overline{\mathbb{Z}}\) (solid, \(p\) not inverted) vs \(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\) / \(\mathbb{R}^{\triangleright}\) (liquid, \(p\) invertible / archimedean) — determines the exactness theorem that applies.
- AB5 failure is the point: the non-exactness of \(\mathrm{Cond}(\mathrm{Ab})\) is not a bug to be hidden; it is the measure-theoretic content that the solid/liquid completion is designed to resolve.
Refs¶
- Clausen–Scholze, Condensed Mathematics (lecture notes, 2019)
- Clausen–Scholze, Analytic (lecture notes, 2019/2020)
- Barwick–Haine, Pyknotic Mathematics (preprint)
- Scholze, Lectures on Analytic Geometry (lectures, Tübingen 2019)
Next¶
Foundations · Solid Modules · Analytic Rings · Pyknotic & Applications · Topos theory · Functional analysis · Derived AG · Prismatic cohomology