Condensed Mathematics: Foundations¶
Job: define condensed sets as sheaves on the pro-étale site of a point (equivalently sheaves on profinite sets for the effective-epimorphism topology), establish that \(\mathrm{Cond}(\mathrm{Set})\) is a Grothendieck topos and \(\mathrm{Cond}(\mathrm{Ab})\) is abelian with enough projectives, and state the exactness obstruction (Clausen–Scholze) that motivates the solid/liquid refinements.
Core¶
Condensed sets¶
Fix a (strongly inaccessible) cardinal \(\kappa\) and let \(\mathrm{Prof}_{\kappa}\) be the category of \(\kappa\)-small profinite sets (= limits of finite sets of cardinal \(<\kappa\)). Endow \(\mathrm{Prof}_{\kappa}\) with the effective-epimorphism topology: a family \(\{S_i\to S\}\) covers iff the induced map \(\bigsqcup S_i\to S\) is an effective epimorphism (topologically surjective). A condensed set is a sheaf
and \(\mathrm{Cond}(\mathrm{Set}):=\mathbf{Sh}(\mathrm{Prof}_{\kappa},\,\text{eff.epi.})\) is a Grothendieck topos (in the universe fixed by \(\kappa\)). Equivalently, condensed sets are sheaves on the pro-étale site of a point: \(\mathrm{Prof}_{\kappa}\) is the full subcategory of the pro-étale site of \(\mathrm{Spec}\,\mathbb{C}\) (or any geometric point) on the connected objects.
| Classical notion | Condensed counterpart |
|---|---|
| Topological space \(T\) | condensed set \(S\mapsto \mathrm{Hom}_{\mathrm{top}}(S,T)\) (defined for "reasonable" \(T\)) |
| Compactly generated weak Hausdorff (CGWH) | full subcategory of \(\mathrm{Cond}(\mathrm{Set})\); the functor is fully faithful on CGWH |
| Discrete set \(A\) | constant condensed set \(\underline{A}\), \(S\mapsto A\) (continuous maps \(S\to A\) are locally constant) |
| Point \(\ast\) | representable \(\mathbb{Z}[-]\)-unit; the final condensed set |
| Profinite set \(S\) itself | representable sheaf \(T\mapsto\mathrm{Hom}_{\mathrm{Prof}}(T,S)\) |
| Topological abelian group \(G\) | condensed abelian group \(S\mapsto\mathrm{Hom}_{\mathrm{top}}(S,G)\) (when \(G\) is "condensible": compactly generated, etc.) |
Comparison with CGWH (Clausen–Scholze). The functor \(\mathrm{CGWH}\to\mathrm{Cond}(\mathrm{Set})\), \(T\mapsto(S\mapsto\mathrm{Hom}(S,T))\), is full and faithful. It is not an equivalence: there are condensed sets not arising from a CGWH space (e.g. the quotient of \(\mathbb{R}/\mathbb{Z}\) "as a condensed set" with the quotient topology differs from the representable of the topological quotient in pathological ways on non-ED test sets), and the obstruction is precisely that \(\mathrm{CGWH}\) is not a topos (no subobject classifier) whereas \(\mathrm{Cond}(\mathrm{Set})\) is.
Condensed abelian groups¶
\(\mathrm{Cond}(\mathrm{Ab}):=\mathrm{Ab}(\mathrm{Cond}(\mathrm{Set}))\) is the category of abelian group objects in the topos, equivalently sheaves of abelian groups on \(\mathrm{Prof}_{\kappa}\). The representables \(\mathbb{Z}[S]\) (free condensed abelian group on the profinite set \(S\)) form a set of projective generators: \(\mathrm{Hom}(\mathbb{Z}[S],M)=M(S)\), and \(\mathbb{Z}[S]\) is projective because \(S\) is projective in the site (profinite sets are projective objects for the eff-epi topology iff they are extremally disconnected, ED).
| Theorem (Clausen–Scholze, Condensed Mathematics) | Statement |
|---|---|
| \(\mathrm{Cond}(\mathrm{Set})\) is a topos | \(\mathbf{Sh}(\mathrm{Prof},\text{eff.epi.})\); has all limits/colimits, subobject classifier |
| \(\mathrm{Cond}(\mathrm{Ab})\) is abelian | kernels and cokernels computed as sheaves; abelian category with all colimits |
| Enough projectives | \(\{\mathbb{Z}[S]\mid S\text{ extremally disconnected}\}\) are projective generators |
| Ext = sheaf-Ext | \(\mathrm{Ext}^i_{\mathrm{Cond}}(M,N)=R^i\mathrm{Hom}_{\mathrm{Cond}}(M,N)\) computed by a projective resolution by \(\mathbb{Z}[S]\) with \(S\) ED |
| Non-exactness | \(\mathrm{Cond}(\mathrm{Ab})\) is not AB5: filtered colimits are not exact; equivalently \(\mathrm{Ext}^1_{\mathrm{Cond}}(\mathbb{R},\mathbb{Z})\neq 0\) |
The exactness obstruction¶
Filtered colimits in \(\mathrm{Cond}(\mathrm{Ab})\) are computed objectwise on \(\mathrm{Prof}\), then sheafified. Sheafification is left exact but not exact on the eff-epi topology: the failure is measured by \(\varprojlim^1\) of towers indexed by non-extremally-disconnected profinite sets. The witness is
where \(\mathbb{R}\), \(\mathbb{Z}\) carry the condensed topology induced by the classical topological groups. This non-vanishing is the measure-theoretic content (it detects the failure of a category of "topological abelian groups" to support a clean homological algebra for functional analysis) and is the obstruction the solid/liquid completion is built to kill. It is not a defect of \(\mathrm{Cond}(\mathrm{Ab})\) as a category — it is the correct category — but a signal that one should pass to the left-completed subcategory of solid modules to recover AB5-style exactness.
Solid modules (pointer)¶
Define \(\mathbb{Z}[S]^{\blacksquare}:=\varprojlim_n \mathbb{Z}[S_n]\) for \(S=\varprojlim S_n\) with \(S_n\) finite (the solid completion of the free condensed abelian group). A condensed abelian group \(M\) is solid if \(\mathrm{Hom}(\mathbb{Z}[S]^{\blacksquare},M)\to M(S)\) is an isomorphism for all profinite \(S\); equivalently \(M\) is a module over the analytic ring \(\overline{\mathbb{Z}}\). The category \(\mathrm{Solid}_{\mathbb{Z}}\) is an abelian subcategory, stable \(\infty\)-category, and closed under \(\otimes_{\blacksquare}\), all limits/colimits — and is AB5. This is the engine of Solid Modules; the abelian-category fix is the content of Analytic Rings.
See it¶
\(\mathrm{Ext}^1_{\mathrm{Cond}}(\mathbb{R},\mathbb{Z})\neq 0\), and why it kills classical topological abelian groups. Consider the short exact sequence of condensed abelian groups
In a hypothetical AB5-exact category of "topological abelian groups" one would want \(\mathrm{Ext}^1(\mathbb{R}/\mathbb{Z},\mathbb{Z})\) to be controlled by classical Pontryagin duality; in \(\mathrm{Cond}(\mathrm{Ab})\) one computes via an ED-projective resolution and finds a non-trivial class — concretely, the class of the extension above splits in the solid category (where \(\mathbb{R}\cong\mathbb{R}\otimes_{\blacksquare}\overline{\mathbb{Z}}\) is solid and the solid \(\mathrm{Ext}^1\) vanishes after completion), but not in \(\mathrm{Cond}(\mathrm{Ab})\) itself. The non-vanishing is equivalent to the non-exactness of \(\varprojlim\) on the tower \(\cdots\to\mathbb{Z}/p^3\to\mathbb{Z}/p^2\to\mathbb{Z}/p\) indexed by \(\mathbb{N}^{\mathrm{op}}\) (a non-ED profinite boundary). This single computation is the reason condensed mathematics does not stop at \(\mathrm{Cond}(\mathrm{Ab})\): one is forced into solid/liquid to do functional analysis.
Check¶
Show that \(\mathrm{Cond}(\mathrm{Set})\) has a subobject classifier, and explain why CGWH does not — i.e. why the comparison functor cannot be an equivalence.
In any sheaf topos \(\mathbf{Sh}(C,J)\) the subobject classifier is \(\Omega(S)=\{\text{closed sieves on }S\}\). For \(\mathrm{Prof}\) with the eff-epi topology, \(\Omega\) is the condensed set of "closed \(J\)-ideals"; a mono \(A\hookrightarrow B\) of condensed sets is classified by \(B\to\Omega\). CGWH has no subobject classifier: a subobject of a CGWH space need not be CGWH (the inclusion is a cofibration, not a general mono), and the lattice of open/closed subobjects does not assemble into a CGWH object representing all monos. Hence \(\mathrm{Cond}(\mathrm{Set})\not\simeq\mathrm{CGWH}\); the comparison is full faithful but not essentially surjective.
Prove that \(\mathbb{Z}[S]\) is projective in \(\mathrm{Cond}(\mathrm{Ab})\) iff \(S\) is extremally disconnected, and use this to explain the role of ED sets in the projective generator.
A condensed abelian group \(P\) is projective iff \(\mathrm{Hom}(P,-)\) preserves epimorphisms, i.e. iff every surjection \(M\twoheadrightarrow N\) induces a surjection \(M(S)\twoheadrightarrow N(S)\) for all \(S\) with \(P=\mathbb{Z}[S]\). Surjections of condensed sets are locally split over covers; the lifting problem for \(S\) is exactly the lifting of maps \(S\to N\) to \(M\), which is the projectivity of \(S\) in \(\mathrm{Prof}\) for the eff-epi topology. Projective objects of \(\mathrm{Prof}\) are exactly the extremally disconnected profinite sets (Stone duality: ED \(\Leftrightarrow\) projective Boolean algebra). Hence \(\mathbb{Z}[S]\) projective iff \(S\) ED, and \(\{\mathbb{Z}[S]\mid S\text{ ED}\}\) is the canonical projective generator set — the ED hypothesis is forced, not optional.
Pitfalls¶
- \(\mathrm{Cond}(\mathrm{Set})\neq\mathrm{CGWH}\): the comparison is full faithful on a subcategory, not an equivalence; quotients in the two categories differ, and "the condensed set underlying a topological space" forgets nothing reasonable but does not identify the categories.
- Projectives need ED: the free condensed abelian group \(\mathbb{Z}[S]\) is projective only for extremally disconnected \(S\); using arbitrary profinite \(S\) in a "projective resolution" is a category error that produces spurious Ext classes.
- Universe size: one fixes a \(\kappa\) and works in \(\mathrm{Prof}_{\kappa}\); changing \(\kappa\) changes the topos. The pyknotic variant (Pyknotic) is the accessible presentation that avoids universe bookkeeping.
- AB5 failure is not a bug: \(\mathrm{Cond}(\mathrm{Ab})\) is the correct abelian category; the failure of filtered-colimit exactness is the measure-theoretic obstruction that the solid/liquid completion is designed to remove. Do not "repair" \(\mathrm{Cond}(\mathrm{Ab})\) by hand — pass to Solid Modules.
Refs¶
- Clausen–Scholze, Condensed Mathematics (2019), Lectures I–III (condensed sets, abelian groups, the obstruction)
- Scholze, Lectures on Analytic Geometry (Tübingen 2019), §1 (pro-étale site of a point)
- Barwick–Haine, Pyknotic Mathematics §1 (accessible variant; comparison)
- Johnstone, Sketches of an Elephant (topos background; effective-epimorphism topologies)
Next¶
Solid Modules · Analytic Rings · Pyknotic & Applications · Condensed index · Topos theory · Functional analysis · Derived AG · Prismatic cohomology