arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 3 new papers, 7 cross-lists and 10 replacements in Category Theory (math.CT), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We construct an endofunctor on (small) simplicial sets which freely adjoins a zero object to a (small) infinity-category, as the co-cone construction freely adjoins a terminal object. We then prove that this endofunctor induces a left adjoint to the inclusion of the infinity-category of (small) pointed infinity-categories and morphisms that preserve zero objects into the infinity-category of (small) infinity-categories. This adjunction, in turn, leads to the definition of a coherent choice of nullhomotopies on a pointed infinity-category. A coherent choice of nullhomotopies is a non-canonical, coherent choice of "higher zeroes" on a pointed infinity-category, mirroring the obvious definition of higher zeroes on the differential graded nerve of a differential graded category of chain complexes over an additive category. We then show that coherent choices of nullhomotopies give an alternate characterization of fiber sequences in pointed infinity-categories, generalizing the 1-categorical definition of kernels in terms of morphisms that compose to zero.
We characterise the commutative rings of weak global dimension at most one whose derived categories can be stratified by the Balmer-Favi support theory. For such rings, this yields a bijective correspondence between the class of localising subcategories of the derived category, and the subsets of the Zariski spectrum. For a large class of rings of weak global dimension at most one including all valuation domains, this occurs if and only if the ring is locally strongly discrete, if and only if the Telescope conjecture holds, and if and only if the module category has Gabriel dimension. However, we also give an example of a valuation domain where all these conditions fail, and yet the residue fields generate its derived category and we can classify all the localising subcategories of $D(R)$. Additionally, in the same example, we show that the Balchin-Stevenson smashing support cannot stratify $D(R)$ either.
We study the tensor triangular geometry of derived categories of valuation domains, obtaining complete ring-theoretic characterizations of their main structural properties. In particular, although valuation domains are almost always non-noetherian, their derived categories always satisfy the local-to-global principle and hence the detection property. We establish precise criteria for homological detection, stratification, and homological stratification, revealing a clear hierarchy among these properties. Notably, we obtain examples of derived categories of commutative rings that are homologically stratified and satisfy the local-to-global principle, but are not stratified. We also determine when the category of modules over a valuation domain has Gabriel dimension, and use this to provide a negative answer to a question of Balchin, G\'omez and Stevenson. Overall, valuation domains provide a rich class of non-noetherian rings whose derived categories can be understood in considerable depth.
We extend the categorical treatment of Bayesian filters to models with feedback, whose transitions and observations may depend on the past observations. We work in the stochastic category, whose objects are measurable spaces and whose morphisms are Markov kernels, which we refer to as stochastic morphisms. We systematically parameterize stochastic morphisms by a measurable space, and develop the calculus of conditioning in a notation intuitively compatible with classical probability theory, requiring neither string diagrams nor densities. A stochastic morphism into a product space is regarded as a parameterized family of joint laws, and its conditional law is again a stochastic morphism, so that the rules of conditioning become equalities of stochastic morphisms. We define the Bayesian inverse with respect to a parameterized prior and prove Bayesian coherence. A model with feedback is then a model whose parameter space grows with the observations, and we derive the Bayesian filter, prediction and smoothing for such models, with standard Borel hidden states and countably generated observation spaces. The affine Gaussian stochastic morphisms parameterized by a measurable space form a family strictly larger than the Markov category $\mathbf{Gauss}$, closed under composition and Bayesian inversion, in which we obtain the Kalman filter and the Rauch-Tung-Striebel smoother with feedback.
Let $\La_1$ and $\La_2$ be finite-dimensional $\ta_2$-finite complex algebras of global dimension at most two. We prove that, if they are cluster equivalent, then they admit $\bbZ$-gradings for which the bounded derived categories of graded modules are triangle equivalent. The equivalence is not required to commute with the internal grading shifts. The proof uses common multigraded dg models, relative gradings on two-term silting objects, and covering quotients. We also consider tilting lifts of cluster-tilting objects with endomorphism algebras of global dimension at most two. In acyclic cluster type, the existence of such a lift for every cluster-tilting object implies piecewise heredity. We construct a counterexample without this assumption using a surface algebra associated with a one-holed torus.
Given a finitely aligned higher-rank $k$-graph $\Lambda$, Farthing, Muhly and Yeend (Semigroup Forum 2005) constructed an inverse semigroup $S_\Lambda$ to provide a groupoid model for the $C^*$-algebra of $\Lambda$. In this paper, we extend some of the existing results related to the semigroup structure of a graph inverse semigroup to the more general setting of the inverse semigroup $S_\Lambda$. We obtain geometric descriptions of the Green's relations on $S_\Lambda$, which effectively generalize the existing characterization of Green's relations on a graph inverse semigroup. We provide a simple and useful description of the Cuntz-Krieger ideal of the Booleanization of $S_\Lambda$ and, based on this, identify a certain $X$-to-join Booleanization of $S_\Lambda$, where the abstract and concrete Cuntz-Krieger relations agree. Besides the natural partial order, $S_\Lambda$ is also ordered by set-inclusion. This helps us to realize $S_\Lambda$ as a semigroup \emph{graded-ordered} by $\mathbb{Z}^k$ and, subsequently, gives rise to a full inverse subsemigroup $S_\Lambda^h$ of $S_\Lambda$, which is graded by $\mathbb{Z}^k$ in the usual sense. We show that the Kumjian-Pask algebra of $\Lambda$ is graded isomorphic to the quotient of the contracted semigroup algebra of $S_\Lambda^h$ by the tight ideal. In the final part of the paper, we show that a $k$-graph morphism $p:\Lambda_1\longrightarrow \Lambda_2$ that is injective and satisfies a certain path lifting property induces, in a natural way, an injective $0$-preserving semigroup homomorphism from $S_{\Lambda_1}$ to $S_{\Lambda_2}$. In view of this, we build a suitable categorical setup to conclude that the isomorphism established between the Kumjian-Pask algebra of $\Lambda$ and the tight ideal quotient of the semigroup algebra of $S_\Lambda^h$ is natural.
Tree decompositions describe large graphs as gluings of smaller component parts. Here we study their categorical duals, tree co-decompositions, which construct large graphs via fibered products of small constituent parts. Co-decompositions are compressed representations of graphs or relational structures: the decomposed object can be exponentially larger than the space required to specify the co-decomposition. Our main contribution is an algorithmic meta-theorem for first order logic: given only a co-decomposition of a $\sigma$-structure $G$ with $b$ bags, each of size at most $w$ and an FO sentence $\phi$, there is an algorithm which, without ever constructing or inspecting $G$, operates on the co-decomposition and determines whether $G \models \phi$ in time $f_\sigma(w,|\varphi|)\cdot b$. We pair this algorithm with structural results on tree co-width, including its relationship to tree-width and a separation from twin-width. Furthermore, we characterize bounded path co-width using a graph grammar. In contrast to our logical meta-theorem, we prove that $(s,t)$-Path becomes NP-hard on co-decomposition inputs. These results demonstrate how categorical constructions can enable efficient computation on exponentially compressed structures.
Let $R$ be a commutative noetherian ring. We ask whether complexes with finitely generated cohomology and a fixed support can be generated by relatively compact objects, using coproducts which exist in the half-bounded derived category. We prove that $\mathbf D^+_{\mathrm{fg},W}(R)=\operatorname{Loc}_{\mathrm{ex}}(\mathbf D^b_{\mathrm{fg},W}(R))$ whenever $W\subseteq\operatorname{Spec} R$ is specialization closed and has finite dimension. The same equality holds for every specialization-closed subset of $\operatorname{Spec} B$ when $B$ is a finite-type algebra over a noetherian domain $A$ of finite character and either $A$ or $B$ is regular. These equalities classify the thick Hom-submodules closed under existing coproducts. We also prove the corresponding bounded-above statement from the support results of Matsui--Takahashi. For an arbitrary noetherian ring, the bounded-below generation conjecture reduces to torsion complexes over quotient domains, with their support unchanged. Finally, over a regular noetherian domain of infinite dimension, we give a bounded-below object which is generated by bounded objects with the same support but has no bounded-below representative by finitely generated modules. The comparison functor from the bounded-below derived category of finite modules is therefore neither full nor essentially surjective.
Three constructions of vertex operator algebras realizing the Haagerup center were announced independently and almost simultaneously. The relations between these constructions are studied in this paper. The generators are expressed in a common lattice basis for $E_6\oplus A_2$. It is proved that the constructions of Riesen and Huang give isomorphic vertex operator algebras, and that this algebra and the algebra constructed by Jiao have isomorphic fixed point subalgebras under automorphisms of order two. The isomorphisms are given explicitly on generators. The common fixed point subalgebra contains the parafermion vertex operator algebras $K(\mathfrak{sl}_2,3)$ and $K(\mathfrak{sl}_2,13)$. A general rationality criterion for unitary vertex operator algebras is given in terms of positive linear combinations of conformal vectors and is applied to these constructions. The two vertex operator algebras are simple current extensions of the common fixed point subalgebra and have equivalent module categories as ribbon categories. The equivalence follows from decomposition with respect to a nondegenerate pointed subcategory with four simple objects in the module category of the common fixed point subalgebra. It is also proved that the conformal nets associated with these unitary vertex operator algebras are completely rational.
Using Mac Lane's PROPs, one can introduce multilinear polynomial identities that involve all the operations of a Hopf algebra: the (co)multiplication, the (co)unit and the antipode. We find bases for polynomial identities and calculate their codimensions for the Sweedler algebra, treating it first as an ordinary associative algebra over a field and second as a Hopf algebra.