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Lectures on Condensed Mathematics

Published 5 May 2026 in math.NT, math.AG, math.CT, math.FA, and math.GN | (2605.03658v1)

Abstract: This is an updated version of the lectures notes for a course on condensed mathematics taught in the summer term 2019 at the University of Bonn. The material presented is joint work with Dustin Clausen. This is intended as a stable citable version of the original lectures, with mostly cosmetic changes to the original document, together with some small corrections.

Authors (1)

Summary

  • The paper develops a sheaf-theoretic framework that recasts topological algebra into condensed abelian groups, resolving classical issues with exactness and tensor products.
  • The study establishes key equivalences between sheaf conditions on compact spaces and coherent duality, enabling explicit computations in derived categories.
  • Key results include the definition of solid abelian groups and a comprehensive six-functor formalism, advancing applications in arithmetic and analytic geometry.

Authoritative Summary of "Lectures on Condensed Mathematics" (2605.03658)

Motivation and Foundational Problems

The document systematically develops the theory of condensed mathematics, framing it as an approach to resolve deep foundational issues arising in algebraic structures endowed with a topology. Notably, classical categories such as topological abelian groups fail to satisfy Grothendieck's axioms for abelian categories, obstructing long exact sequences and base change in cohomology. Furthermore, the interaction between topological structures and derivations (e.g., continuous group cohomology) often fails to preserve categorical exactness. Traditional completions of tensor products (especially of topological modules) are ambiguous, yielding divergent practices across contexts. Condensed mathematics, originating from Scholze and Clausen, proposes a robust framework that addresses these issues via sheaf-theoretic and categorical constructions on the pro-étale site and profinite sets.

Condensed Sets and Abelian Groups

The central idea is the recasting of topological objects (sets, groups, rings) as sheaves on a site of profinite sets. The category of condensed sets, $\Cond(\mathcal C)$ for a category C\mathcal C, is defined as the category of C\mathcal C-valued sheaves on the pro-etale site of a point. These sheaves satisfy a strict set of conditions ensuring compatibility with disjoint unions and surjective maps (corresponding to finite covers in the site).

A foundational result is that the category of condensed abelian groups forms an abelian category satisfying Grothendieck's axioms (AB3–AB6, AB3*, AB4*). Key structural properties are:

  • Existence of compact projective generators: These generators arise from extremally disconnected spaces (e.g., Stone–Čech compactifications), allowing for expressive and exact resolutions.
  • Stability under limits and colimits: The category is closed under arbitrary products, direct sums, and exact filtered colimits.
  • Equivalence of sheaf conditions on compact Hausdorff and extremally disconnected spaces: This equivalence enables explicit computations and functorial resolutions.

Moreover, condensed mathematics provides a symmetric monoidal tensor product and internal Hom objects, permitting enriched categorical and homological algebra.

Cohomology and Derived Categories

Cohomology in condensed mathematics respects traditional topological invariants while resolving pathologies of classical topological group cohomology. Specifically:

  • For any compact Hausdorff space SS, condensed cohomology Hcondi(S,Z)H^i_{\text{cond}}(S,\mathbb Z) coincides with sheaf cohomology Hsheafi(S,Z)H^i_{\text{sheaf}}(S,\mathbb Z) and Čech cohomology, as established in Theorem 3.11. Furthermore, for profinite sets, higher cohomology vanishes and H0H^0 corresponds to continuous functions.
  • The derived category $D(\Cond(\Ab))$ is shown to be compactly generated, and all RR-functors between objects (including locally compact abelian groups) admit explicit computation. The theory utilizes a functorial resolution of abelian groups (Breen–Deligne resolution), ensuring control of derived functors and Ext groups in the condensed setting.

A strong claim is the full faithfulness of the functor from the bounded derived category of locally compact abelian groups $D^b(\LCA)$ to C\mathcal C0, guaranteeing that categorical and homological constructions in classical topological contexts are faithfully represented.

Solid Abelian Groups and Completed Tensor Products

A major advance is the definition and characterization of solid abelian groups as a subclass of condensed abelian groups, designed to formalize completion with respect to measures on profinite sets. Solid abelian groups are those for which every map from a profinite set C\mathcal C1 extends uniquely to measures, enabling integration and completion in a nonarchimedean sense.

Key results:

  • Solidification functor: The inclusion of solid abelian groups into condensed abelian groups admits a colimit-preserving left adjoint, producing a symmetric monoidal "completed" tensor product C\mathcal C2 and its derived version C\mathcal C3.
  • Tensor product property: For arbitrary products of copies of C\mathcal C4, the completed derived tensor product yields products indexed by C\mathcal C5, i.e., C\mathcal C6.
  • Pathological completions become exact: Pathologies in classical tensor products (e.g., C\mathcal C7) are eliminated or clarified via the solidification procedure.

The theory extends to analytic rings and modules, providing a general framework for measure-theoretical completions (solid modules, analytic rings) that preserve categorical exactness and enable strict functoriality.

Globalization and Coherent Duality

The final lectures develop a globalization procedure, constructing the derived category of solid modules on discrete adic spaces (SpaC\mathcal C8) and demonstrating descent for sheaves of C\mathcal C9-categories. This enables gluing of local categories of solid modules to coherent global objects, crucial for applications in arithmetic geometry and C\mathcal C0-adic Hodge theory.

The six-functor formalism is constructed in this context: pullbacks, pushforwards, C\mathcal C1, C\mathcal C2, tensor products, and internal Hom, all respecting the condensed structures. Explicit computations demonstrate that for smooth, separated maps C\mathcal C3 of finite type, the dualizing complex is canonically identified as C\mathcal C4, with functorial trace and compatibility with classical Serre–Grothendieck duality.

The functor C\mathcal C5 defined via the adic space formalism preserves compact objects, providing a new proof of finiteness of coherent cohomology for proper maps, and enabling the full six-functor formalism in solid (condensed) settings.

Implications and Directions

Condensed mathematics establishes a well-behaved, functorial framework for algebraic and analytic geometry with topological structures. It resolves foundational obstructions in abelian categories, tensor products, and cohomology for topological groups and analytic rings. The categorical properties (compact projective generators, exactness of colimits, symmetric monoidal structure) enable systematic, explicit computations of derived functors between complex algebraic structures.

Practical implications include:

  • Fully justified cohomological and duality formalisms for non-discrete rings, C\mathcal C6-adic analytic spaces, and adic spaces.
  • Extension of classical results (e.g., Serre duality) to non-proper and non-archimedean settings, with canonical constructions and explicit computational recipes.
  • Application to arithmetic geometry and the theory of locally compact abelian groups, including derived computations and sheaf-theoretic descent.

Theoretical ramifications suggest further developments in the six-functor formalism at the C\mathcal C7-categorical level, enhanced duality theorems, and broader applicability to completed tensor products in analytic settings. Potential future directions include deep integration with set theory and forcing (as hinted by connections to extremally disconnected spaces), as well as structural advances in presentable big toposes and higher categorical algebra.

Conclusion

"Lectures on Condensed Mathematics" (2605.03658) provides a rigorous, comprehensive structure for reconciling topology with algebra through sheaf theory on profinite sites. It results in abelian and derived categories with favorable exactness, compactness, and tensor properties. The framework resolves classical pathologies, supports explicit cohomological computations, and furnishes a universal categorical language for advanced applications in geometry, topology, and arithmetic.

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Overview

This paper (really a set of lecture notes) introduces “condensed mathematics,” a new way to do algebra and geometry when your objects carry a topology (a notion of closeness). The big idea is to replace raw topological spaces by “condensed” objects that remember all the continuous information you care about, but live in a setting where algebra works smoothly. This lets you take kernels, cokernels, tensor products, and do homological algebra (long exact sequences, derived categories) without the usual headaches that occur with topological groups and vector spaces.

The main questions

The lectures are motivated by simple-sounding but surprisingly tricky questions:

  • How can we do algebra (like adding, taking kernels, forming quotients) with objects that have a topology, such as topological groups, rings, or vector spaces?
  • How can we make cohomology (a way to measure “holes” or “global patterns”) behave well for topological groups and spaces?
  • How can we build a solid foundation for analytic geometry over topological fields (like the real numbers or the p-adic numbers), where continuity matters?

Classically, these goals run into problems: topological abelian groups don’t form a “nice” algebraic universe, short exact sequences don’t give the expected long exact sequences in continuous cohomology, and completed tensor products are hard to set up uniformly.

The approach, in everyday terms

Think of a complicated object (like a topological group) as something you can “probe” with many simple, well-behaved test shapes. If you know what all continuous maps from these test shapes into your object look like, you can reconstruct the object’s continuous behavior.

  • The test shapes: very nice compact spaces called “profinite sets” (imagine big, tidy, zero-dimensional compact spaces) and, even more special ones called “extremally disconnected” spaces (super-tidy: separating pieces is as easy as possible).
  • The probes: for each test shape S, collect all continuous maps S → T into your object T. Do this for every S, in a way that “glues” correctly when you cover S by smaller pieces. This gluing rule is the sheaf condition (like making sure a jigsaw puzzle forms a consistent picture when you fit overlapping pieces together).

This leads to the definition:

  • A condensed set is a rule T that assigns to each profinite S the set T(S) of continuous maps S → T, satisfying the sheaf gluing rules.
  • Similarly, condensed groups, rings, and modules are the same idea but with algebraic structure.

Why this helps: Instead of working directly with topologies (which can misbehave algebraically), you work with all continuous maps from tidy probes. This keeps the continuity information but moves you into a world where algebra (limits, colimits, kernels, cokernels) behaves perfectly.

A few technical aids make this safe and precise:

  • Size control: to avoid set-theory paradoxes, we temporarily restrict to test shapes below a very large “size limit” κ (a strong limit cardinal). Later, the notes show the theory doesn’t depend on which κ you choose.
  • Equivalent viewpoints: you can probe with all compact Hausdorff spaces, or just with extremally disconnected ones; both give the same condensed objects. The extremally disconnected ones give especially clean algebra.

An important bridge back to topology:

  • There is a faithful way to send any (reasonable) topological space X to a condensed set underline{X} by recording all continuous maps from probes S into X.
  • For sufficiently nice spaces (compactly generated ones), this embedding is fully faithful, and there is a left adjoint that recovers a natural topology on the underlying set of a condensed object. Think of this as “remembering just enough topology to go back and forth.”

Main findings and why they matter

Here are the core results the lectures establish and explain why they’re big deals.

  • Condensed abelian groups form an excellent abelian category. In simple terms: you can add them, form kernels and cokernels, take exact sequences, and all the standard tools of homological algebra work just as in the familiar world of plain abelian groups. Even better, this category satisfies powerful exactness properties (traditionally labeled AB3, AB4, AB5, AB6, and their duals), which guarantee that big constructions like products, direct sums, and filtered colimits preserve exactness. This is precisely what fails for topological abelian groups in the usual sense.
  • There are compact projective generators. That means there are especially nice “LEGO brick” objects from which every condensed abelian group can be built, and which make computations (like derived functors) actually doable. These bricks come from extremally disconnected spaces via free abelian constructions.
  • Tensor products and internal Homs exist and behave well. You can form M ⊗ N and Hom(M, N) inside the condensed world, and they satisfy the expected adjunctions. Free condensed abelian groups Z[T] are flat, so tensoring behaves as nicely as in ordinary algebra.
  • Derived categories are well-behaved. Because there are enough projectives and the category is so nice, you get a robust derived category D(Cond(Ab)) with derived tensor and derived Hom. This unlocks long exact sequences and spectral sequences in contexts that involve topology.
  • The condensed viewpoint aligns with classical topology for nice spaces. For compact Hausdorff spaces, and more generally compactly generated weak Hausdorff spaces, you can faithfully move back and forth between the topological and condensed pictures. That means condensed mathematics doesn’t throw away classical information; it organizes it so algebra works.

A telling example: consider the map from the real numbers with the discrete topology to the real numbers with the usual topology. In the ordinary topological category, the “difference” between these is hard to package algebraically. In the condensed world, the difference is an honest condensed abelian group Q. Its underlying set looks trivial, but on profinite test shapes S it captures the genuinely new continuous data. This shows how condensed objects can “see” structure that naive approaches miss.

Implications and impact

  • A unified, reliable foundation for doing algebra with topology. Many theories that blend algebra and continuity (representation theory of topological groups, analytic geometry over R or Q_p, continuous cohomology) become cleaner. Exactness issues vanish; long exact sequences and derived functors behave as expected.
  • Better tools for modern geometry and number theory. Condensed mathematics underlies new developments in analytic geometry and stacks, and interacts well with perfectoid methods and p-adic Hodge theory.
  • Clearer connections to logic and set theory. Extremally disconnected spaces also appear in forcing and models of set theory, hinting at deep structural links. The notes briefly explain how these worlds touch.
  • Practical computations become feasible. Having compact projective generators and a well-behaved tensor/derived setup means many complicated constructions can be handled systematically.

In short, these lectures present a framework that keeps all the continuous information we care about, but moves it into a mathematically “well-lit laboratory” where the usual algebraic machinery works flawlessly. This opens the door to tackling problems across topology, algebra, and arithmetic with a common, powerful toolkit.

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Knowledge Gaps

Knowledge gaps, limitations, and open questions

The text establishes foundational aspects of condensed sets and condensed abelian groups, but it leaves several conceptual, structural, and computational questions unresolved. The following list highlights concrete gaps and directions where further work is needed:

  • Precise characterization of which sheaves on the pro-étale site of a point are “condensed”: beyond the sufficient conditions given (e.g., T1 spaces with quasicompact maps from points), give necessary and sufficient conditions for a sheaf on profinite (or compact Hausdorff) sets to be the left Kan extension of its restriction to κ-small extremally disconnected (ED) sets for some uncountable strong limit κ.
  • Non-T1 spaces are not captured by condensed sets via the functor X ↦ 𝑋̲: develop an alternative “condensed-like” framework (or a modified site/topology) that accommodates non-separated spaces (e.g., Sierpiński-type) while preserving desirable exactness and derived properties.
  • Underlying-topology functor T ↦ T(∗)ₜₒₚ is not symmetric monoidal and need not preserve products; for condensed groups/rings, T(∗)ₜₒₚ may fail to carry a compatible topological group/ring structure. Identify:
    • Conditions on a condensed group/ring ensuring its underlying-topology is compatible with the algebraic structure.
    • A refined “topologization” functor that preserves finite products (or is symmetric monoidal), or a criterion ensuring product-preservation on a useful subcategory.
  • Change of cardinal κ: the functor Cond_κ → Cond_{κ′} is fully faithful and preserves all colimits and λ-small limits (λ = cf(κ)) in the cases treated, but:
    • Determine minimal hypotheses on a general category C (beyond a conservative forgetful functor to Sets) under which change-of-κ preserves all small limits and retains exactness properties.
    • Exhibit counterexamples or prove optimality of the λ-small limit restriction in broader settings (e.g., non-algebraic C).
  • Big topos limitation: the category of condensed sets is not the category of sheaves on any small site and lacks a set of generators. Develop:
    • A small-site or presentable substitute that is equivalent on a large subcategory of interest (e.g., quasiseparated objects), or
    • A systematic “big topos” toolkit (beyond [Brink]) for standard topos-theoretic constructions (e.g., internal logic, small object arguments) in this setting.
  • Heavy reliance on extremally disconnected (ED) spaces:
    • Since ED spaces are rare and not closed under products, clarify how this affects constructions that use products at the site level, and quantify the impact on monoidal and internal Hom structures.
    • Develop computational models of generators (or alternative sites) that avoid explicit use of Stone–Čech compactifications while retaining compact projective generators and AB6/AB4*.
  • Reflection to topological spaces: characterize the “accidental identification” phenomenon when passing from quasiseparated condensed sets T to T(∗)ₜₒₚ in compactly generated weak Hausdorff (CGWH) spaces.
    • Give necessary and sufficient conditions on filtered diagrams of compact Hausdorff spaces under which the reflection is fully faithful, and identify minimal enhancements of CGWH spaces that make the reflection faithful on a prescribed class of colimits.
  • Complete comparison with pyknotic sets: fill in a precise, fully functorial equivalence between (universe-dependent) pyknotic sets and condensed sets, including:
    • The translation of AB6, compact projective generators, and internal Hom/tensor structures.
    • Stability of exactness properties across the equivalence and the impact on analytic stacks.
  • Cohomology of condensed objects:
    • Provide detailed proofs and generalizations of comparison results (e.g., vanishing statements such as H¹_{cond}(S, ℝ_{disc}) = 0 for profinite S and comparisons with singular/Čech cohomology for compact Hausdorff spaces), and extend to non-compact spaces.
    • Establish functorial long exact sequences in condensed cohomology mirroring classical continuous group cohomology and identify the precise hypotheses needed for derived functors to behave well.
  • Analytic geometry and completed tensor products:
    • Bridge the gap between the sheaf-theoretic tensor ⊗ on Cond(Ab) and classical completed tensor products in functional analysis and analytic geometry.
    • Give criteria ensuring that M ⊗ N (or M ⊗ᶫ N) recovers standard completions for Banach, Fréchet, or locally convex spaces; spell out base-change and flatness results needed for quasicoherent sheaves over condensed rings.
  • Condensed rings and modules:
    • Extend AB6/AB4* and compact-projective-generator statements from Cond(Ab) to module categories over general condensed rings; characterize when condensed rings admit enough compact projective modules and the consequences for the compact generation of derived module categories.
    • Analyze exactness of filtered colimits and tensor products in Cond(R-Mod) for non-discrete condensed rings.
  • Forcing and set-theoretic dependence:
    • Systematically analyze which properties of condensed categories are absolute under forcing extensions (beyond the remarks on ED spaces and forcing), and identify invariants sensitive to changes of the ground model.
    • Explore whether forcing-inspired constructions yield new generators, sites, or structural theorems in condensed mathematics.
  • Constructive control of sizes and sheafification:
    • Provide explicit bounds and constructive procedures for the sheafification of presheaves like S ↦ ℤ[T(S)] on the ED site, including effective control of the Stone–Čech surjections and their sizes in terms of |S|.
    • Develop practical algorithms for computing colimits and sheafification without passing to very large ED covers when possible.
  • Internal Hom and limits/colimits:
    • Give concrete descriptions of (M, N)(S) for basic condensed groups (e.g., 𝑋̲ where X is compact Hausdorff or discrete), and identify general conditions under which internal Hom preserves filtered colimits or satisfies adjointness with derived tensor at the level of D(Cond(Ab)).
  • Cardinal-profile variants:
    • Beyond the brief note on κ regular (e.g., “light condensed sets” for κ = ℵ₁), develop a systematic theory of how categorical and exactness properties vary with regular vs. singular κ and identify the trade-offs (e.g., which limits/colimits are preserved) most useful for applications such as analytic stacks.
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Practical Applications

Summary

This paper introduces condensed sets (and, in particular, condensed abelian groups) as sheaves on the pro-étale site of a point (modeled by profinite or, equivalently, extremally disconnected compact Hausdorff spaces). It resolves long-standing foundational issues with doing algebra over topological rings/groups/modules by moving to an abelian, Grothendieck-style category with compact projective generators, internal Hom, tensor products, and a well-behaved derived category. It also clarifies how classical topological objects (e.g., compact Hausdorff spaces) embed and are recovered in the condensed world, and how cohomology can be computed coherently. These features enable more robust definitions, proofs, and computations across areas that involve topology and algebra simultaneously.

Below are actionable applications grouped by deployment horizon, with sector links and feasibility notes.

Immediate Applications

These can be adopted now by researchers, educators, and tool builders using the paper’s constructions as-is.

  • Academia (Mathematics; topology/geometry/algebra)
    • Replace topological abelian groups with condensed abelian groups to restore abelian-category tools
    • What: Model topological modules/groups as objects in Cond(Ab) to get exact products, filtered-colimit exactness, long exact sequences in cohomology, and compact projective generators.
    • Where: Continuous group cohomology; representation theory over topological groups (e.g., GLn(R), GLn(Qp)); sheaf-theoretic settings in topology/geometry.
    • Tools/Workflows: Use internal Hom and symmetric monoidal tensor on Cond(Ab); compute derived functors in D(Cond(Ab)) via projective resolutions.
    • Assumptions/Dependencies: Restrict to κ-compactly generated or T1 spaces when passing to/from topology; familiarity with sheafification on extremally disconnected sites; handling of universes/large cardinals (or use pyknotic sets as an alternative).
    • Uniform, sheaf-theoretic base-change and completion in analytic contexts
    • What: Interpret base change M ⊗̂A B via sheafification/tensoring in Cond(Ab) to avoid ad hoc topological completions.
    • Where: Analytic geometry over topological fields (e.g., R, Qp); relative situations in derived/analytic geometry.
    • Tools/Workflows: Perform base change as sheafified tensor M ⊗ N in Cond(Ab); leverage exactness properties to preserve short exact sequences and derived functors.
    • Assumptions/Dependencies: Availability of condensed models for the analytic objects in use; translation of existing Banach/locally convex structures into condensed modules where needed.
    • Robust long exact sequences in continuous cohomology
    • What: Compute continuous group cohomology in an abelian/derived setting that guarantees long exact sequences.
    • Where: Galois/étale/representation-theoretic cohomology with topological coefficients.
    • Tools/Workflows: Use D(Cond(Ab)) with compact projectives; cohomology via RHom in Cond(Ab).
    • Assumptions/Dependencies: Set up of condensed module structures for the coefficients; replacement of classical “continuous cochains” with condensed resolutions.
    • Cohomology of compact Hausdorff spaces via condensed methods
    • What: Reinterpret singular/sheaf cohomology for compact Hausdorff spaces using their qcqs condensed avatars.
    • Where: Algebraic topology; comparison theorems; derived functor computations on compact spaces.
    • Tools/Workflows: Identify compact Hausdorff X with a qcqs condensed set; compute cohomology via sheaf/cochain complexes in Cond(Ab).
    • Assumptions/Dependencies: Use of the equivalence between compact Hausdorff spaces and qcqs condensed sets; standard homological algebra in Cond(Ab).
  • Software/Tooling (Computer algebra, theorem proving, category theory libraries)
    • Formalization of condensed categories and derived functors
    • What: Implement Cond(Set), Cond(Ab), internal Hom, tensor, and D(Cond(Ab)) in proof assistants/CAS.
    • Where: Lean/Coq/Agda and category-theory libraries; homological-algebra toolkits.
    • Tools/Workflows: Provide AB axioms, compact projective generators, exactness of products/filtered colimits; offer projective resolutions in Cond(Ab).
    • Assumptions/Dependencies: Universe management (large cardinals or pyknotic alternative); performance considerations for sheaves on extremally disconnected sites.
    • Safer APIs for topological algebra
    • What: Expose condensed-abstraction layers instead of ad hoc topological categories that fail exactness.
    • Where: CAS/DSLs used by researchers in topology/geometry/number theory.
    • Tools/Workflows: Standardize tensor/RHom in Cond(Ab); ensure long exact sequences are machine-checkable.
    • Assumptions/Dependencies: Migration plan from classical topological-abelian-group APIs to condensed counterparts.
  • Education (Graduate teaching; expository materials)
    • Curriculum updates leveraging condensed foundations
    • What: Teach topological algebra, cohomology, and compactly generated spaces via condensed sets, clarifying exactness and derived constructions.
    • Where: Graduate courses in algebraic topology, functional analysis over topological fields, arithmetic geometry.
    • Tools/Workflows: Use κ-compactly generated spaces ↔ condensed sets; emphasize extremally disconnected sites for intuition.
    • Assumptions/Dependencies: Audience familiarity with basic sheaf theory and category theory; choice of universe framework (e.g., pyknotic sets for set-theoretic convenience).
  • Logic/Set Theory (Foundations; forcing pedagogy)
    • Sheaf-theoretic view of forcing via extremally disconnected spaces
    • What: Use the paper’s equivalences to illustrate forcing extensions as categories of sheaves on extremally disconnected compact spaces.
    • Where: Logic courses; expositions connecting condensed mathematics and forcing.
    • Tools/Workflows: Construct Shv∧(S) for extremally disconnected S; compare to pyknotic/condensed approaches.
    • Assumptions/Dependencies: Comfort with forcing/ETCSR; careful cardinal/size management.

Long-Term Applications

These require further theory-building, scaling, or integration across ecosystems.

  • Software/Scientific Computing (Symbolic/derived computation; p-adic/analytic algorithms)
    • Computation of derived invariants in analytic/arithmetical settings using condensed frameworks
    • What: Develop algorithms for p-adic/analytic cohomology and base change within Cond(Ab) or condensed module categories.
    • Where: Arithmetic geometry toolchains; computation in p-adic Hodge theory; spectral sequences in condensed categories.
    • Tools/Products: Libraries for derived tensor/RHom over condensed rings; high-level APIs for cohomology of sheaves/complexes on condensed spaces.
    • Assumptions/Dependencies: Mature condensed models for analytic objects (e.g., condensed Banach modules); optimization for large derived computations.
    • Integration of condensed mathematics into mainstream CAS and proof assistants
    • What: Comprehensive libraries covering Cond(Set/Ab), condensed modules over topological/analytic rings, and their derived categories.
    • Where: Lean/Coq math libraries; Sage/Magma-style systems for arithmetic geometry.
    • Tools/Products: Certified derived-functor engines; automation for long exact sequences; interfaces to extremally disconnected-site sheaves.
    • Assumptions/Dependencies: Community consensus on universes (condensed vs. pyknotic); sustained maintenance and performance engineering.
  • Academia (Advanced geometry/representation theory)
    • Condensed analytic geometry and stacks as default foundations
    • What: Build analytic stacks and geometry over topological fields using condensed objects to simplify base change, completions, and derived operations.
    • Where: Non-archimedean/archimedean analytic geometry; geometric representation theory; relative Langlands-style settings.
    • Tools/Workflows: Derived categories of condensed sheaves; six-functor formalisms in condensed contexts; compact generation for cohomological control.
    • Assumptions/Dependencies: Extension of the present foundations to broader analytic categories; compatibility with existing geometric frameworks.
    • Unified framework for continuous representations
    • What: Model unitary/smooth representations over R/Qp via condensed modules, enabling cleaner homological algebra and deformation theory.
    • Where: p-adic and real representation theory; topological group actions in derived settings.
    • Tools/Workflows: Derived deformation functors in Cond(Ab); condensed-module spectral sequences; continuity handled via the site.
    • Assumptions/Dependencies: Detailed development of “condensed” functional-analytic structures; consensus on replacing classical topological vector spaces.
  • Education/Community Standards
    • Standardization of condensed methods in graduate curricula and textbooks
    • What: Transition to condensed mathematics as the standard for algebra over topological structures.
    • Where: Algebraic topology, number theory, arithmetic geometry programs.
    • Tools/Products: Textbooks, lecture notes, and problem sets grounded in Cond(Ab) and D(Cond(Ab)).
    • Assumptions/Dependencies: Broad instructor familiarity; availability of accessible expositions and exercises.
  • Logic/Foundations
    • Cross-pollination with forcing and big topos theory
    • What: Develop unified categorical frameworks connecting big topoi (condensed) and forcing models for foundational research and pedagogy.
    • Where: Category-theoretic set theory; independence results; logical frameworks for large-scale mathematics.
    • Tools/Workflows: Big-presentable-category tooling; categorical forcing interfaces.
    • Assumptions/Dependencies: Continued research on big topoi and their computational/axiomatic handling.

Notes on Feasibility and Dependencies

  • Set-theoretic size issues: While the paper uses uncountable strong limit cardinals κ to control sizes, it also provides κ-independence mechanisms and alternatives (e.g., pyknotic sets). Implementations must choose a consistent universe policy.
  • Topology interface: The condensed/topological adjunction works best for T1 and κ-compactly generated spaces; non-separated spaces (e.g., Sierpiński) fall outside the condensed image without additional hypotheses.
  • Extremally disconnected site: Many constructions are phrased via sheaves on extremally disconnected spaces. While conceptually powerful, concrete computations may require specialized data structures and optimizations.
  • Cultural/transition costs: Adopting condensed mathematics in established workflows (analytical/topological) requires retooling and training; near-term gains are strongest where exactness/derived behavior previously caused friction.
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Glossary

  • Abelian category: An additive category where every morphism has a kernel and cokernel and all monomorphisms/epimorphisms are normal, enabling exact sequences. "The category of condensed abelian groups is an abelian category which satisfies Grothendieck's axioms (AB3), (AB4), (AB5), (AB6), (AB3*) and (AB4*), to wit: all limits (AB3*) and colimits (AB3) exist, arbitrary products (AB4*), arbitrary direct sums (AB4) and filtered colimits (AB5) are exact, and (AB6) for any index set JJ and filtered categories IjI_j, j∈Jj\in J, with functors i↦Mii\mapsto M_i from IjI_j to condensed abelian groups, the natural map"
  • Adjoint functor theorem: A foundational result guaranteeing existence of adjoints under completeness/cocompleteness and smallness conditions. "To prove the existence of compact projective generators, we use that by the adjoint functor theorem, the forgetful functor from κ\kappa-condensed abelian groups to κ\kappa-condensed sets has a left adjoint T↦Z[T]T\mapsto \mathbb Z[T]."
  • Big topos: A “large” topos-like category that need not be sheaves on a small site but retains many topos properties. "This has been studied in detail by Brink, \cite{Brink}, under the term big topos (and big presentable categories)."
  • Beth number: A hierarchy of infinite cardinalities defined by transfinite recursion, often written ℶα\beth_\alpha. "For example, define ℶα\beth_\alpha inductively for all ordinals α\alpha via ℶ0=ℵ0\beth_0=\aleph_0, ℶα+=2ℶα\beth_{\alpha^+} = 2^{\beth_\alpha} for a successor ordinal and for a limit ordinal as the union of all smaller ℶα\beth_{\alpha}'s."
  • Cofinality: The smallest cardinality of a cofinal subset in a partially ordered set; for a cardinal, the minimal size of an unbounded subset. "The functor T↦Tκ′T\mapsto T_{\kappa^\prime} is fully faithful and commutes with all colimits and all λ\lambda-small limits where λ\lambda is the cofinality of κ\kappa."
  • Compact projective object: An object PP such that Hom(P,−)(P,-) preserves filtered colimits (compact) and epimorphisms lift along maps from PP (projective). "Moreover, the category of κ\kappa-condensed abelian groups is generated by compact projective objects."
  • Compactly generated space: A space where continuity can be tested on maps from compact Hausdorff spaces. "Recall that a topological space XX is compactly generated if a map f:X→Yf: X\to Y to another topological space YY is continuous as soon as the composite S→X→YS\to X\to Y is continuous for all compact Hausdorff spaces SS mapping to XX."
  • Condensed abelian group: A sheaf of abelian groups on the pro-étale site of a point (or equivalently a functor on extremally disconnected sets satisfying finite-product compatibilities). "In fact, condensed abelian groups form an abelian category of the nicest possible sort."
  • Condensed set: A sheaf of sets on the pro-étale site of a point, encoding topological information via maps from profinite sets. "A condensed set is a sheaf of sets on $\ast_$. Similarly, a condensed ring/group/... is a sheaf of rings/groups/... on ∗.\ast_."
  • Derived category: A triangulated category obtained from chain complexes by inverting quasi-isomorphisms, used to do homological algebra. "As $\Cond(\Ab)$ has enough projectives, one can form the derived category $D(\Cond(\Ab))$."
  • Equalizer: The limit of two parallel morphisms, capturing the subset on which they agree. "Then the value T(S)T(S) is determined as the equalizer of the two maps $T(\tilde{S})\to T(\tilde{\tilde{S})$."
  • Extremally disconnected (space): A compact Hausdorff space where every continuous surjection onto it splits. "A compact Hausdorff space SS is extremally disconnected if any surjection S′→SS^\prime\to S from a compact Hausdorff space splits."
  • Filtered colimit: A colimit over a filtered category, often used to model “direct limits” of systems with coherent transition maps. "arbitrary direct sums (AB4) and filtered colimits (AB5) are exact"
  • Forcing extension: A new model of set theory obtained via forcing, adding sets to a ground model. "Namely, any extremally disconnected compact Hausdorff space can be used to construct a forcing extension of the ground model, and conversely any forcing extension arises this way."
  • Fully faithful functor: A functor inducing bijections on Hom-sets; an embedding of categories. "This functor admits a natural left adjoint, which is a functor from κ\kappa-condensed sets to κ′\kappa^\prime-condensed sets. This functor is fully faithful and commutes with all colimits and many limits (in particular, all finite limits), as we will prove in Proposition~\ref{prop:changekappa} below."
  • Grothendieck's axioms (AB3, AB4, AB5, AB6, AB3*, AB4*): Exactness and (co)completeness properties characterizing well-behaved abelian categories. "The category of condensed abelian groups is an abelian category which satisfies Grothendieck's axioms (AB3), (AB4), (AB5), (AB6), (AB3*) and (AB4*), to wit: all limits (AB3*) and colimits (AB3) exist, arbitrary products (AB4*), arbitrary direct sums (AB4) and filtered colimits (AB5) are exact, and (AB6) ..."
  • Ind-compact Hausdorff space: An object expressible as a filtered colimit of compact Hausdorff spaces, typically with closed immersion transition maps. "Quasiseparated condensed sets are arguably better behaved: In fact, they are equivalent to the category of ind-compact Hausdorff spaces lim→⁡iXi\varinjlim_i X_i where all transition maps are closed immersions."
  • Internal Hom: An object representing morphisms in a closed monoidal category, enabling Hom to be an object rather than just a set. "for any condensed abelian groups MM, NN, the group of homomorphisms $\Hom(M,N)$ has a natural enrichment to a condensed abelian group, defining an internal $\Hom$-functor object (M,N)(M,N)."
  • Kan extension (left Kan extension): A universal way to extend a functor along another functor; left Kan extensions preserve colimits. "Under this equivalence, the functor T↦Tκ′T\mapsto T_{\kappa^\prime} corresponds to left Kan extension along the full inclusion of κ\kappa-small extremally disconnected sets into κ′\kappa^\prime-small extremally disconnected sets."
  • Metrizable space: A topological space whose topology arises from a metric. "Any first-countable topological space XX (in particular, any metrizable topological space) is compactly generated; in fact κ\kappa-compactly generated for any uncountable κ\kappa."
  • Pro-étale site: A Grothendieck topology built from projective limits of étale maps, here specialized to a point. "The pro- " etale site $\ast_$ of a point is the category of profinite sets SS, with finite families of jointly surjective maps as covers."
  • Profinite set: A compact, totally disconnected, Hausdorff space that is an inverse limit of finite discrete sets. "The pro- " etale site $\ast_$ of a point is the category of profinite sets SS, with finite families of jointly surjective maps as covers."
  • Pyknotic sets: A closely related framework to condensed sets using different set-theoretic conventions (universes and tiny profinite sets). "they term these pyknotic sets."
  • QCQS (quasi-compact and quasi-separated): A finiteness and separation condition (here for condensed sets) mirroring the scheme-theoretic notion. "The functor X↦X‾X\mapsto \underline{X} induces an equivalence between compact Hausdorff spaces XX and qcqs condensed sets TT."
  • Quasicompact: A property analogous to compactness for non-Hausdorff settings; every open cover has a finite subcover (or a map/object is “quasicompact” in the condensed sense). "then R=S×TS⊂S×SR=S\times_T S\subset S\times S is a quasicompact sub-condensed set,"
  • Quasiseparated: A separation condition ensuring fiber products behave well; in schemes/condensed sets, intersections of qc opens are qc. "A compactly generated space XX is weak Hausdorff if and only if X‾\underline{X} is quasiseparated."
  • Quotient map: A surjective continuous map inducing the quotient topology on the codomain. "any surjection S′→SS^\prime\to S of compact Hausdorff spaces is a quotient map"
  • Quotient topology: The finest topology on a target set making a surjection continuous; sets are open iff their preimage is open. "the underlying set equipped with the quotient topology for the map ⊔S→XS→X\sqcup_{S\to X} S\to X"
  • Sheaf: A functor satisfying gluing conditions on a site, encoding local-to-global data. "A condensed set is a sheaf of sets on $\ast_$. Similarly, a condensed ring/group/... is a sheaf of rings/groups/... on ∗.\ast_."
  • Sheaf condition: The exactness conditions ensuring sections over a cover can be uniquely glued from compatible local sections. "satisfying T(∅)=∗T(\emptyset) = \ast and the following two conditions (which are equivalent to the sheaf condition):"
  • Sheafification: The process of turning a presheaf into a sheaf by enforcing the sheaf condition. "We note that a priori QQ is the sheafification of this functor, but it turns out that sheafification is unnecessary,"
  • Sierpinski space: The two-point topological space with one open singleton, a standard non-T1 example. "The problem are nonseparated spaces like the Sierpinski space X={s,η}X=\{s,\eta\} where {η}\{\eta\} is open but {s}\{s\} is not."
  • Site: A category equipped with a Grothendieck topology (covers) on which sheaves are defined. "The site ∗κ-\ast_{\kappa\text-} is the site of κ\kappa-small profinite sets SS with covers given by finite families of jointly surjective maps."
  • Stone–Čech compactification: The universal compact Hausdorff compactification of a completely regular space; for a discrete set, a profinite Stone space of ultrafilters. "for example by taking the Stone--\v{C}ech compactification of SS considered as a discrete set."
  • Strong limit cardinal: A cardinal κ\kappa such that 2λ<κ2^\lambda<\kappa for all λ<κ\lambda<\kappa; often assumed uncountable here. "In these lectures, we will always assume that κ\kappa is an uncountable strong limit cardinal, i.e.~κ\kappa is uncountable and for all λ<κ\lambda<\kappa, also 2λ<κ2^\lambda<\kappa."
  • Symmetric monoidal tensor product: A bifunctor ⊗ with associativity/commutativity constraints giving a monoidal structure. "It has a symmetric monoidal tensor product −⊗−-\otimes -, where M⊗NM\otimes N is the sheafification of S↦M(S)⊗N(S)S\mapsto M(S)\otimes N(S)."
  • T1 space: A topological space where all singletons are closed (Frechet-separable axiom). "If XX is a topological space all of whose points are closed (i.e.~XX is T1T1), then X‾\underline{X} is a condensed set for which all maps from points are quasicompact."
  • Ultrafilter: A maximal filter on a set; points of the Stone space of a Boolean algebra. "as κ\kappa is a strong limit cardinal and S′S^\prime is the set of ultrafilters on the set SS, i.e.~a subset of the powerset of the powerset of SS."
  • Ultraproduct: A construction combining structures modulo an ultrafilter, yielding a limit-like object. "If SS is a Stone--\v{C}ech compactification, this is simply an ultraproduct of the category Set\mathrm{Set},"
  • Weak Hausdorff space: A compactly generated space where images of maps from compact Hausdorff spaces are closed Hausdorff subspaces. "A compactly generated space XX is weak Hausdorff if and only if X‾\underline{X} is quasiseparated."
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