One crossed module unifies three braided-category setups
A single definition covers group-graded, group-action, and G-crossed braided categories, with a cohomological classification.
math.CT
Category Theory
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra
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The prime at infinity is a blurred interval
Re-examined through topos theory, the real place is a non-Hausdorff interval that the classical picture misses.
2-categories of abelian and triangulated categories are 2-homological
The 2-catgorical framework shos that Serre quotients and Verdier localizations are istances of 2-cokernels.
· “Exactness of the 2-categories of abelian and triangulated categories”
Atomically generated ideals unify telescope conjectures
A new atomic smashing frame generalises the Balmer spectrum to any symmetric monoidal ∞-category, with applications to chromatic homotopy…
2-coherence fixes left-adjoint colimit preservation in HoTT
A new coherence condition ensures suspension and modalities preserve graph-indexed colimits in homotopy type theory.
· “On Left Adjoints Preserving Colimits in Homotopy Type Theory”
Geometric field theory moduli become equivariant mapping spaces
Main theorem expresses d-dimensional theory spaces as maps of O(d)-equivariant smooth presheaves on Cartesian spaces.
Model structure makes cartesian 2-fibrations as strict as 2-functors
Marked anodyne extensions and a path-object construction reduce the weak coherence of 2-fibrations to a combinatorial homotopy theory.
Nine axioms pin down expression tree categories
Every free non‑symmetric operad's matching category is captured by a short list of conditions, and gains a noetherian form akin to group…
Local Ocneanu rigidity forces 2^{dim} bounds on separable algebra structures
Every semisimple Hopf algebra has at most 2^{dim H-1} left coideal subalgebras; irreducible II1 subfactors get 2^{[M:N]-1}.
Every ergodic cubeset is a tower of abelian bundles
A categorical proof of the weak structure theorem that also covers condensed cubesets and compact nilspaces.
One theorem makes Frobenius and Cartier modules dual
The two long-studied module categories are exact opposites, and previous dualities appear as special cases.
A finite poset cut builds an equivalent 'negative' twin
The admissible cuts of a poset yield derived-equivalent partner posets, and the transformation reverses cleanly.
· “Derived equivalences between diagram categories of finite posets”
One t-norm condition decides when fuzzy numbers form a domain
Full completeness only under Gödel logic; domain and approximation structure under (S).
Arrow notation maps Euclid's ratios without algebra
A diagrammatic syntax drawn directly on the Elements' diagrams makes Book V and VI arguments visually legible and philosophically faithful.
· “A Categorical Approach to Euclidean Ratios and Proportions”
Four labels say when time data survives a viewpoint switch
A sheaf framework classifies persistent versus cumulative views and extends to tree-width and multi-agent networks.
· “Time-Varying Data as Sheaves: an Invitation to Narratives”
Explicit pushout for strict symmetric monoidal categories
The construction preserves central idempotents, enabling models of concurrent resource exchange.
One Bell probability settles loop consistency: it must equal 1/d²
Postselected output then equals the categorical trace, tying non-causal logic to teleportation.
· “Tracing the Loop: Non-Causal Computation, Partial Traces, & Postselected Entanglement”
Obstruction class decides when a pre-triangulated category is triangulated
A cohomological criterion reduces Verdier's octahedral axiom to a vanising condition, with applications to exotic categories and parameter…
· “From Pre-triangulation to Triangulation: Obstructions and Exact Lifting”
Auslander–Reiten conjecture proven for radical-cube-zero algebras
A single finite bound on self-extensions guarantees projectivity for all such algebras of Loewy length at most three.
· “The Auslander-Reiten conjecture for algebras with radical cube zero”
No injective embeddings for cosheaves on non-semi-separated schemes
The obstruction is a locally cotorsion cosheaf with no map into any locally injective one.
· “The contraherent version of the theorem of Slavik and Stovicek”
One completeness condition on Frobenius pairs yields Hovey triples
A single assumption unifies and extends many Gorenstein model structures in relative homological algebra.
· “Cotorsion pairs and model structures induced by resolution dimensions relative to Frobenius pairs”
Monoidal derived equivalences force categorical equivalence
Classification of bounded monoidal t-structures by universal grading characters reconstructs semisimple tensor and module categories.
· “Derived Reconstruction of Semisimple Tensor and Module Categories”
Category theorists prove a version of the snake lemma
Proves a Snake Lemma for 2-categories with a strong bizero object under 2-di-exactness or non-self-dual hypotheses.
Third rung: triples alone detect arithmetical categories
One pushout-limit condition on three regular epimorphisms captures congruence-distributive categories and re-proves Pixley.
· “A new characterization of arithmetical categories and Pixley's theorem”
Differential cubical forms define a Cartan calculus in tangent categories
A single construction recovers de Rham forms on manifolds, schemes, and diffeological spaces, bridging two known algebraic extremes.
Only the minimum t-norm makes [0,1]*-Set cartesian closed
The classification of cartesian closed real-valued set categories extends from continuous to all left-continuous t-norms, with only one…
· “Cartesian closedness of the category of real-valued sets, II”
Representing categories as numbers and matrices
A framework shows that finite sets, vector spaces, and relations become executable categories in MATLAB/Octave.
· “From Matrices to Morphisms II: Examples of Computational Categories in MATLAB and Octave”
Weak Lie3- groups for 2- gerbes on torus fibrations reduce to T- duality
Constructing a commutative diagram of 3- groups that dimensionally reduce to the known T- duality 2- groups, promising higher T- duality.
· “Weak Lie 3-groups, 2-gerbes over torus fibrations of type F1, and T-Duality”
One framework unifies all twisted Hochschild invariants
The paper discovers that for any bicategory B equipped with G-twisting data (a strong monoidal functor Θ_A: G → Pic_B(A) for each object…
Three invariants classify every spectral duality structure
Combinatorial type (k,f), degree K, and inversion classes {r, 1/r} determine each structure and its response rank.
Galois connection pairs subalgebras with tensor subcategories
Order-reversing maps between two lattices become a lattice anti-isomorphism on intervals fixed by explicit closure operators.
· “A Galois connection between subalgebras and tensor subcategories”
A balanced companion is unique
It also pins down when one exists: exactly when the saturated objects are closed under limits and directed colimits.
· “Beth companions of finitary essentially algebraic theories”
Residue characteristic decides when Artin motives stratify
A full support-theoretic classification for motive categories, with the telescope conjecture as a corollary.
Nerves of Steel conjecture gets a new counterexample
Two morphisms with zero tensor product and no nilpotence break the homological-to-prime comparison map.
· “Another counterexample to the Nerves of Steel Conjecture”
Classical restriction categories get a unique override operator
The paper axiomatizes override and update for partial maps and shows every classical restriction category admits exactly one such operation.
Lie correspondence built from tangent structure alone
Every Lie group in a tangent category yields an internal differential Lie algebra, recovering classical theories.
· “The Lie group-Lie algebra correspondence in tangent categories”
Presentation functor fails to preserve tilting property
A concrete 4-vertex path algebra provides the first counterexample to Yang's Question 3.15.
· “A negative answer to a question on tilting objects and two-term complexes”
Dependent-type structures outrun generalised categories with families
They match a broader 'higher comprehension' framework, and a ring example shows cases the old one cannot reach.
· “(2-dep,Sigma)-categories are not generalised categories with families”
New theory links braided differentials to quantum G-structures
It proves that quantum G-structures are reductions of quantum frame bundles, generalising Cartan geometry.
· “On Braided Differential Calculi and Quantum G-structures”
Sheaf categories inherit the diffeology-topology adjunction
Even with no map between their sites, sheaves on a space and on its D-topology get exact adjoint functors.
· “An adjunction of the categories of sheaves related to a topology and a diffeology”
Octahedral axiom fails for faithful noninvertible suspension
Construction turns a pre-triangulated counterexample into a right one while preserving the obstruction to the octahedral axiom.
· “A right pretriangulated category which is not right triangulated”
One radical condition restores Hopkins–Levitzki for groupoid-graded rings
Artinian plus objectwise-nilpotent radical forces noetherianity, and semiprimary rings make graded artinian equal noetherian.
· “Hopkins-Levitzki Type Theorems for Groupoid Graded Rings”
Spec Z's archimedean place is an absolute twistor line
Construction over F_{1^2} yields an equivariant topos whose complex points carry the fixed-point-free twistor involution, completing the…
· “The Absolute Twistor Line and the Geometry of overline{Spec\, mathbf Z}”
Integral profiles classify finite-dimensional modules completely
Two modules match exactly when their parameterized integrals are similar, tying homological dimensions to polynomials.
· “Integral coefficient rings and homological dimensions of algebras”
Stable ∞-categories are oplax limits of their semiorthogonal pieces
A reconstruction theorm turns semiorthogonal decompositions into gluing data, with an application to Beilson's collection on projective…
· “Semiorthogonal decompositions of stable infty-categories”
Ryū Tomonaga proves that higher cluster-tilting categories inside differential graded…
d-stable dg categories are exactly coherent connective dg categories of weak global dimension at most d+1 with a shifted duality, and any…
· “Auslander correspondence for higher stable dg categories and cluster Morita theory”
Same theorem makes four logic-algebra categories coextensive
A single characterisation of coextensive morphisms in bounded hoops covers Wajsberg, join-hoop, Heyting and MV-algebra categories.
This paper builds two new model structures for dagger (∞,1)-categories
Dagger (∞,1)-categories get two new model structures, on dagger simplicial categories and dagger simplicial sets, connected by a Quillen…
· “Models for dagger (infty,1)-categories I: Dagger simplicial sets and unitary cores”
Right ARS duality survives every relative theory
The dualizing functor is exact and fully faithful on ideal quotients, preserving almost split sequences and the AR formula.
One rule now describes all functors into higher span categories
The same construction models spans at every dimension, up to infinity, enabling higher Hall algebras.
A module-pair recipe builds model structures on Q-shaped modules
The same lift that recovers chain-complex homotopy theory also builds model structures on differential modules.
Fiber data plus transport decide the total factorization system
Ambifibrations straighten to double functors; the forgetful functor to categories is monadic.
· “Factorization Systems on infty-Categories: Un/Straightening and Monadicity”
Explicit pre-9-angulated category that is not 9-angulated
Cubing a twisted comparison over the two-element field keeps the obstruction, so the higher mapping-cone axiom fails.
· “A pre-(n+2)-angulated category which is not (n+2)-angulated”
Two products on symmetric sequences obey coherence law
A proof of a 2014 conjecture shows that composition and matrix products satisfy an interchange, making every operad a Day-operad.
Strongly bounded objects get bounded t-structures only when already bounded
Finite strong finitistic dimension forces strongly bounded subcategory to equal bounded subcategory for a bounded t-structure to exist
· “Bounded t-structures on the category of strongly bounded objects”
Non-contractible, yet every finite truncation is contractible
Every finite truncation of this nerve collapses; only the whole object remembers the difference.
· “About the contractibility of the walking coinductive equivalence”
Twisted symplectic flag classes are independent in Grothendieck ring
Noncommutative motives prove linear and quadratic independence, giving strong structural constraints
· “Independence of the Grothendieck classes of twisted symplectic flag varieties”
Thin flat surfaces reduce to one canonical normal form
Normal form plus sufficient relations gives a complete generators-and-relations presentation of TFS.
Simple singularities are fixed by their singularity categories
The AR quiver alone is blind; the Tyurina algebra steps in and distinguishes the singularities.
· “Singularity Categories of Simple Singularities in Positive Characteristic”
Model structure for dependent-type theories bridges syntax and semantics
A Quillen model structure on contextual categories shows that the homotopy theory of gats is that of display map categories, with a…
· “A categorical model structure for generalized algebraic theories”
Power monoids are Kleisli convolution monoids
A natural isomorphism turns power monoids into convolution monoids, yielding a categorical proof of the Tringali–Yan rigidity conjecture.
Module categories carry all higher Morita equivalences of E2-algebras
For n=1,2, the module-realization functor is an equivalence in the finite setting, tying algebraic Morita theory to topological defects.
Free Heyting algebra on two generators ruled out as topos truth-value lattice
Settles decades-old problem in categorical logic: not every Heyting algebra can be the subterminal-object lattice of an elementary topos.
· “Failure of Higher-Order Truth within Intuitionistic Propositional Logic”
New condition solves even‑n case of Auslander–Reiten angle problem
A non‑cancellation condition restores local finiteness from Grothendieck group relations in even‑dimensional higher angulated categories.
Every quantale is a formal concept lattice
One Galois-closed relation between a semigroup and a set now encodes a quantale, homomorphisms included.
Finitistic dimensions unified under tilting generators
Several definitions coincide; recollement yields equivalence of finiteness across categories, generalizing ring theory.
· “Finitistic dimensions in triangulated categories with a compact silting generator”
Category dilatation theory is now fully machine-checked
A proof assistant verifies the universal property, the key operations, and two corrections to the earlier printed theory.
Natural numbers object is projective in the free topos
Categorical proof recovers Makkai's argument establishing countable choice for intuitionistic higher-order logic
· “Makkai's lost proof of projectivity of N in the free topos”
Every dual adjunction comes from one bifold algebra
A new biequivalence reduces all algebraic dual adjunctions to homming into a single object with two commuting structures.
Every dualizable additive category is an almost-module category
The characterization links dualizable additive categories to idempotent ideals, nuclear modules, and algebraic K-theory.
Every n-cotorsion pair yields a higher abelian heart
One framework covers n-extended t-structure hearts and cluster-tilting quotients, no algebraicity assumption required.
· “Extended heart construction (I): The heart of n-cotorsion pairs on triangulated categories”
Koszul duality is one equivalence of 2-categories
A Morita category of operads and bimodules matches a dual category of cooperads and bicomodules.
Localic Esakia duality becomes fully constructive
Heyting frames meet Esakia locales through conic frames, with no prime ideal theorem.
Three conditions collapse: Baer
For rigid tt-categories, idempotent annihilator ideals, the second tt-De Morgan law, and extremally disconnected dual spectra are…
Comonads from group actions classify groups exactly
This paper proves that a single comonad on sets recovers the full isomorphism type of a group.
Pretorsion theories now need only pullbacks and pushouts
A near-torsion theory always contains a largest subcategory where the classical pretorsion axioms hold.
Group center controls the derived center
For p-adic Lie groups, every central symmetry of compact mod-p representations comes from the group itself, up to nilpotents.
Stone duality is computable for quasi-Polish spaces
Overt discrete and compact Hausdorff spaces encode each other with computable functors and adjunctions
· “Effective quasi-Polish categories of overt discrete spaces and compact Hausdorff spaces”
Every dimension quotient is a boundary limit of presentations
A categorical construction over free presentations recovers the group-ring quotients, with a matching Lie-ring version.
· “Dimension quotients as boundary limits: the general case”
Semi-orthogonal decompositions realized by module subcategories
On a gentle algebra's marked surface, rotating arc systems turns module pieces into derived-category decompositions.
· “Semi-orthogonal and derived decompositions for gentle algebras”
