Papers

Bicategorical shadows provide a categorical framework that encompasses Hochschild homology and topological Hochschild homology (THH), encodes their Morita invariance, and extends the notion of trace from symmetric monoidal categories to bicategories. Certain equivariant variants, such as $C_n$-twisted THH, do not fit into the ordinary shadow framework. We introduce bicategories with $G$-twisting data and show that every ordinary shadow on such a bicategory induces, for each $g\in G$, a $g$-twisted shadow on the associated bicategory of $G$-twists. These $g$-twisted shadows are invariant under $G$-Morita equivalence; examples include $C_n$-twisted THH and twisted Hochschild homology of $C_n$-Green functors. We further define a $g$-twisted bicategorical trace that recovers the $g$-twisted Hattori–Stallings trace. For $C_n$-Green functors, the orbitwise $g$-twisted Hattori–Stallings traces assemble into a morphism of $C_n$-Mackey functors that, at the orbit $C_n/C_n$, recovers the degree-zero twisted Dennis trace.

This paper establishes the algebraic and categorical foundations of spectral duality structures (SDS). We show that the class of all SDS admits the structure of a graded symmetric monoidal category, where the degree $K$ is the fundamental invariant indexing the strata. We define operations of Cartesian product and disjoint union, morphisms preserving involution and weights, and prove that $C^\ast=1/(1+\sqrt K)$ is a functor constant on each stratum. We classify objects up to isomorphism by the combinatorial type $(k,f)$, the degree $K$, and a multiset of inversion classes $[r]=\{r,r^{-1}\}$. In each degree, skeletons are exactly the finite $C_2$-sets over the additive group of the reals with $C_2$ acting by $t\mapsto -t$, which explains the monoidal structure and identifies the classification with the decomposition into orbits. Furthermore, we prove that the response rank is the rank of an explicit matrix built from the amplitudes and the weights, and hence exactly the number of non-trivial pairs $k$ under the full-rank condition, and we connect the categorical structure with the Fisher–Rao geometry of Paper V. The SDS, originally discovered in stochastic resetting problems, is thus an autonomous mathematical object with a rich algebraic structure and a natural geometric realisation.

Let $\mathcal{B}$ be a braided finite tensor category and let $A$ be a simple commutative algebra in $\mathcal{B}$. We construct an order-reversing Galois connection between subalgebras of $A$ and tensor subcategories of $\mathcal{B}_A$. Let $\mathcal{B}'$ denote the Müger center and set $A':=A\cap\mathcal{B}'$. The closure operators are $B\mapsto\langle B,A'\rangle_{\mathrm{alg}}$ and $\mathcal{E}\mapsto\langle\mathcal{E},\mathcal{B}_A^{\mathrm{loc}}\rangle_{\otimes}$. Thus the closed subalgebras are those containing $A'$, while the closed tensor subcategories are those containing $\mathcal{B}_A^{\mathrm{loc}}$; equivalently, the fixed-point intervals $[A',A]_{\mathrm{alg}}$ and $[\mathcal{B}_A^{\mathrm{loc}},\mathcal{B}_A]_{\otimes}$ are anti-isomorphic as lattices. For a finite tensor category $\mathcal{C}$, the canonical algebra in $\mathcal{Z}(\mathcal{C})$ yields an anti-isomorphism between its subalgebras and tensor subcategories of $\mathcal{C}$. When $\mathcal{B}$ is nondegenerate, Frobenius extensions correspond to unimodular tensor subcategories. For a Hopf algebra in $\mathcal{B}$, the correspondence specializes to an order-preserving bijection between Hopf ideals and normal left coideal subalgebras.

For categories, being balanced (meaning that every arrow that is both epic and monic is an isomorphism) can be regarded as a strong tameness property which plays an important role, for example, in algebra and logic. We study the problem of associating a balanced companion category, called a Beth companion, to a locally finitely presentable category (equivalently, the category of models of a finitary essentially algebraic theory). We show that, if it exists, the Beth companion is unique and can be described in terms of saturated objects. Under additional assumptions, we prove that Beth companions can be computed as orthogonality classes, and admit a syntactic presentation via Gabriel–Ulmer duality. Furthermore, we establish conditions for the transfer of properties from a category to its Beth companion. A number of examples of categories that do or do not admit a Beth companion are provided. In particular, we show that the category of commutative rings does not admit a Beth companion, and employ Pierce sheaf representations to describe the Beth companion of other classes of rings.

Let k be a field of characteristic p>0. We prove that for every p-decomposition group P, the tensor-triangulated category \operatorname{DPerm}(P;k) is stratified and its Balmer spectrum is generically noetherian. As an arithmetic application, we show that for a nonarchimedean local field F with residue characteristic \ell, the category \(\operatorname{DAM}(F;k)\) of derived Artin motives is stratified if and only if \ell\neq p. In the stratified case its Balmer spectrum is generically noetherian; consequently, \operatorname{DAM}(F;k) satisfies the telescope conjecture.

In [B2] Balmer introduced the homological spectrum of a rigid tensor-triangulated category, and had the “nerves of steel” not to conjecture that it was in bijection with the prime spectrum, despite an “avalanche of examples”. Naturally, the statement came to be known as Balmer's Nerves of Steel Conjecture. More recently, in [NVY] the conjecture was stated explicitly and extended to a more general non-symmetric context. In this paper we provide a counterexample to the original suggestion in [B2], so also to the conjecture from [NVY]. A first counterexample was recently constructed in [BHR].

We study the override and update operators on partial functions from the perspective of restriction categories. We propose a definition of override restriction categories, in which both of the operators in question exist. We prove a number of results concerning these operators and their relationship to the structure of the ambient restriction category, as well as relating them to the existing literature on the override and update operators. We provide various examples of override restriction categories and in particular show that every classical restriction category is an override restriction category.

Classic Lie theory establishes a correspondence between Lie groups, which are internal group objects in the category of smooth manifolds, and Lie algebras. An analogous correspondence also exists for group objects in affine schemes. Both smooth manifolds and affine schemes form tangent categories, which provide a categorical context for differential geometry. Therefore, it is natural to ask whether the Lie correspondence can be constructed entirely from the tangent structure. Building on work of Cockett and Schwarz, we develop an internal Lie group-Lie algebra correspondence in tangent categories. We introduce Lie group objects as group objects in a tangent category which admit a tangent space at the unit. For any group object, we construct a Lie functor and show it is tangentially representable exactly when the group object is a Lie group. Representability then yields an internal Lie algebra whose underlying object is the tangent space at the unit. We prove this Lie algebra is a differential Lie algebra, with a bilinear Lie bracket induced by the adjoint representation, and we extend the construction to a functor from Lie groups to differential Lie algebras. Finally, we recover the usual Lie correspondences in both differential and algebraic geometry.

We extend Balmer's tensor-triangular geometry to arbitrary presentably symmetric monoidal $\infty$-categories $\mathcal V$ by introducing the atomic smashing frame. This construction is functorial in $\mathcal V$, recovers the classical Balmer spectrum in the stable compactly-rigidly generated case, and yields a formulation of the telescope conjecture for arbitrary $\mathcal V$. Using dualizable and locally rigid $\infty$-categories, we identify smashing ideals with locally rigid localizations and establish recollement results for smashing frames and atomic smashing frames. As a consequence, we obtain a recollement principle for the telescope conjecture. We also establish descent for the telescope conjecture along several natural classes of coverings. As a principal application to chromatic homotopy theory, we analyze the structure of the smashing frame of $\mathrm{Sp}$. We prove a prime decomposition in terms of the smashing frames of the $p$-local $\infty$-categories $\mathrm{Sp}_{(p)}$ and, at each prime, an embedding into the product of the smashing frames of the monochromatic layers $\mathrm{Sp}_{T(n)}$. In particular, the spatiality of the smashing frame of $\mathrm{Sp}$ reduces entirely to that of $\mathrm{Sp}_{T(n)}$. Beyond the stable setting, we characterize the telescope conjecture for $\infty$-topoi in terms of smashing fields, and for connective module categories and hypercomplete connective sheaves in terms of Pierce-type conditions. Finally, we introduce the Serre smashing frame. Over a connective $\mathbb{E}_\infty$-ring $R$, this frame sits between the atomic and usual smashing frames, providing an intermediate structural layer in the study of the telescope conjecture for the connective $R$-module $\infty$-category.

Let $M$ be a silting object in an idempotent complete algebraic triangulated category $\mathcal T$. Put $B={\rm End}_{\mathcal T}(M)$, and let $\mathbb{P}_M\colon {\rm pr}(M)\to K^{[-1,0]}({\rm proj}B)$ be the presentation functor associated with $M$. It was recently asked whether $\mathbb{P}_M(T)$ must be tilting whenever $T\in{\rm pr}(M)$ is a tilting object. We answer this question in the negative by giving an explicit finite-dimensional example. Namely, for $$ \Lambda=k(1\xrightarrow{\alpha}2\xrightarrow{\beta}3\xrightarrow{\gamma}4)/(\alpha\beta\gamma),$$ we construct a silting object $M\in K^b({\rm proj}\Lambda)$ and a tilting object $T=\Sigma\Lambda\in{\rm pr}(M)$ for which $$ {\rm dim}_k{\rm Hom}_{K^b({\rm proj}B)}\bigl(\mathbb{P}_M(T),\Sigma^{-1}\mathbb{P}_M(T)\bigr)=1.$$ Thus $\mathbb{P}_M(T)$ is a two-term silting complex but not a tilting complex.

The notion of a generalised category with families, introduced by Coraglia and Emmenegger is one of the most general notions introduced to capture categorically the notion of dependent typing. It is shown that these generalised categories with families are biequivalent to comprehension categories. We will show that the notion of a (2-dep,$\Sigma$)-category, an extension of the notion of a (dep,$\Sigma$)-category, introduced by Petrakis, is not biequivalent to generalised categories with families, but instead is equivalent to a direct generalisation of that notion.

We prove that a coaction of a Hopf algebra on another preserving the latter's associative-algebra structure and comultiplication automatically also preserves the counit, antipode, and tensor-interchange map and in fact factors through the coaction dual to an action of an affine group scheme by Hopf-algebra automorphisms. This generalizes and unifies a number of results to the effect that quantum groups have only classical symmetries, due to Kasprzak et al., Budziński-Kasprzak and Brannan et al. The stated automatic classicality follows from the fact that the antipode, counit and tensorand-interchange of a Hopf algebra internal to any symmetric monoidal category are contained in the (non-symmetric) monoidal subcategory generated by the associative-algebra structure and the comultiplication, assuming a weak form of arrow-retraction invariance.

We develop a theory of first order differential calculi in braided monoidal categories and classify braided covariant and bicovariant calculi on braided Hopf algebras. We show that, under certain conditions, bicovariant calculi can be transmuted to braided bicovariant calculi. For Radford–Majid biproducts, we combine bicovariant calculi on a Hopf algebra and braided bicovariant calculi on the corresponding braided Hopf algebra to covariant smash product calculi. The associated Maurer–Cartan form is shown to decompose into a direct sum of the Maurer–Cartan forms of the structure Hopf algebra of the quantum principal bundle and the braided Hopf algebra on the base. Geometrically, this construction realises the quantum affine extension of a given Hopf algebra, and we prove that the resulting quantum principal bundle is equipped with a frame resolution induced by the quantum Maurer–Cartan form. Building on this correspondence, we introduce and develop the notion of quantum $\textrm{G}$-structure, proving that quantum $\textrm{G}$-structures are quantum frame resolutions on the reduction. The theory is illustrated by examples based on transmutations of higher analogues of Sweedler's Hopf algebra and on the braided quantum plane, seen as a Yetter–Drinfeld module of $O_q(\mathrm{GL}_2)$.

There exists an adjoint pair (D, C) of functors between the category of diffeological spaces and that of topological spaces. In this article, by using the functors, we introduce two pairs of functors between the categories of sheaves on a diffeological space and on a topological space. Then, the adjointness of the novel functors up to the functors induced by the unit and counit of C and D is clarified. There is no natural functor between sites obtained by a diffeological space and a topological space. Therefore, the adjunctions on the categories of sheaves are elaborated without applying the general theory of topoi.

Chen, Liu, Lu, and Zhang recently constructed a pretriangulated category with invertible suspension in which Verdier's octahedral axiom fails. We introduce a general enlargement construction for right pretriangulated categories and show that it preserves axioms (RTR1)-(RTR3), while failure of (RTR4) is detected by the forgetful functor. Applied to their type $A_5$ example, the construction yields a right pretriangulated category that is not right triangulated. In this example, the suspension is faithful but not essentially surjective.

We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins–Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided $\Gamma_0$-artinian rings need not be $\Gamma_0$-noetherian, but using our tools we prove that one-sided gr-hereditary $\Gamma_0$-artinian rings, two-sided $\Gamma_0$-artinian rings, and $d$-finitely generated one-sided $\Gamma_0$-artinian rings are $\Gamma_0$-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.

We construct the absolute algebraic geometry of the compactification $\overline{\text{Spec}\, \mathbf Z}$ by amalgamating the affine absolute curve $(\text{Spec}\, \mathbf Z)_{\mathbf{F}_{1}}$ with an archimedean component defined over the signed extension $\mathbf{F}_{1^2}$ of $\mathbf{F}_1$. By adjoining a formal imaginary unit to the absolute projective line, we obtain an equivariant topos endowed with a canonical geometric inversion symmetry, which induces the twistor real structure on its complex points. This archimedean geometry is incorporated into a global absolute curve defined as an internal object of the odd arithmetic topos, dual to the multiplicative monoid of odd positive integers, and governed by the intrinsic Hopf structure of spherical $\mathbf{F}_{1^2}$-algebras. The restriction of the absolute Frobenius action to odd integers is forced arithmetically by the extension of scalars to $\mathbf{F}_{1^2}$. On complex points, the resulting dynamics simultaneously generates the Adams operations and complex conjugation on real Hodge structures. At the categorical level, the odd arithmetic topos originates in the pericyclic category, whose $\lambda$-operations provide a conceptual interpretation of the local factors of geometric L-functions.

We define the integral profiles of all modules and introduce integral coefficient rings ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)$ for all finite-dimensional complex algebras $A$. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules $M\cong N$ are isomorphic if and only if their integral profiles are similar, i.e., $M\cong N$ if and only if $\displaystyle \int M \sim \int N$. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of $A$ (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of $A$ is finite if and only if the embedded integral profile of $\mathrm{top}(A)$ lies in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$; (6) the finitistic dimension of $A$ is finite if and only if each embedded integral profile of $M$ lying in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$ implies that its degree is less than or equal to a fixing integer $d\in\mathbb{N}^+$; (7) we provide two sufficient conditions, such that if a finite-dimensional complex algebra $A$ satisfies one of them, then its finitistic dimension is finite.

We define semiorthogonal decompositions of stable $\infty$-categories of length $n$, extending the theory of semiorthogonal decompositions presented in arXiv:2106.02873. Prior to this, we explicitly construct an equivalence of $\infty$-categories relating Waldhausen diagrams to coherent complexes in stable $\infty$-categories. This allows us to view semiorthogonal decompositions from two different perspectives, each of which has its unique advantages and disadvantages. Under mild conditions, we prove a reconstruction theorem for semiorthogonal decompositions, recovering a stable $\infty$-category as the (op)lax limit of a diagram formed by the subcategories constituting its decomposition. We apply this reconstruction in the case of Beilinson's exceptional collection, to obtain a reconstruction of $D^b(\text{Coh}(\mathbb{P}^n))$.

The notion of $d$-stable dg categories axiomatizes $d$-cluster tilting subcategories of stable dg categories. We establish an Auslander correspondence for $d$-stable dg categories: we characterize the $d$-stability of an additive connective dg category in terms of coherence, weak global dimension, and a duality on finitely presented modules. This gives a homological characterization of $d$-stability and reveals it as a twisted form of $(d+1)$-Calabi–Yau duality. For locally finite connective dg algebras, this interpretation becomes particularly transparent under Koszul duality, where $d$-stability corresponds to a shifted self-injectivity condition on the Koszul dual. Following the constructions of Amiot, Guo and Keller, for a $d$-stable dg category $M$, we introduce its $d$-cluster dg category $\mathcal C_{d,{\rm dg}}(M):=\operatorname{per}_{\rm dg}M/^\mathbb{L}\mathcal D^b_{\rm fp, dg}(M)$. Using our Auslander correspondence, we show that $\mathcal C_{d,{\rm dg}}(M)$ contains $M$ as a $d$-cluster tilting subcategory. In particular, every $d$-stable dg category can be realized as a $d$-cluster tilting subcategory of a stable dg category. We then develop cluster Morita theory: a pretriangulated dg category equipped with a $d$-cluster tilting subcategory $M$ is quasi-equivalent to $\mathcal C_{d,{\rm dg}}(M)$. Thus the connective dg structure of a cluster tilting subcategory determines its ambient dg category up to quasi-equivalence. As an application of cluster Morita theory, we prove a Morita-theoretic variant of Amiot's conjecture. More precisely, we establish a Calabi–Yau correspondence: for a locally finite $d$-stable dg category $M$ over a field, right $(d+1)$-Calabi–Yau structures on $\mathcal D^b_{\rm fp, dg}(M)$ are in bijection with right $d$-Calabi–Yau structures on $\mathcal C_{d,{\rm dg}}(M)$.

The main aim of this note is to provide a characterisation of coextensive morphisms in the category of bounded hoops. This characterisation is then used to show that several categories of bounded hoops are coextensive as categories. Among these are the variety of bounded Wajsberg hoops, or more generally the variety of bounded $\vee$-hoops. Our characterisation also yields the coextensivity of the category $\mathbf{Heyt}$ of Heyting algebras and recovers the known coextensivity of the category $\mathbf{MV}$ of MV-algebras.

We develop simplicial-set models for dagger $(\infty,1)$-categories. We first establish a model structure on dagger simplicial categories whose weak equivalences are local weak equivalences that are essentially surjective up to coherently unitary equivalence. We then construct the dagger Joyal model structure on dagger simplicial sets and prove that the dagger rigidification–nerve adjunction is a Quillen equivalence. We next introduce the unitary core and use it to characterize dagger Joyal equivalences intrinsically. Finally, we compare the dagger Joyal model structure with the DCH anti-involutive Joyal model structure.

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a right Auslander-Reiten-Serre (ARS for short) duality $(\tau, \eta)$ in the sense of Iyama, Nakaoka and Palu. We show $(\tau, \eta)$ induces right ARS dualities on the relative theories of $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Under relative structures, we show that the functor $\tau$ is indeed an exact functor between extriangulated categories and preserves almost split exangles. Finally, we give some applications and examples on these results. For instance, we generalize a recent result by A. Hubery to a categorical framework.

In this paper, we construct the $(\infty,\infty)$-category $\mathsf{Span}_\infty(\mathcal{C})$ of spans, also known as correspondences, in any given $(\infty,1)$-category $\mathcal{C}$ with finite limits. This yields new models for the span $(\infty,n)$-categories for $n \in \mathbb{N} \cup \{\infty\}$. We characterize the mapping $(\infty, n-1)$-categories in these $(\infty,n)$-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span $(\infty,n)$-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where "higher" refers to both higher categorical and "higher monoidal" structures, i.e., $\mathsf{E}_k$-algebras in $(\infty, n)$-categories for $n, k>1$.

We develop a method for constructing abelian model structures on the category Q,AMod of Q-shaped modules from cotorsion pairs in AMod, where Q is a small preadditive category satisfying certain conditions and AMod denotes the category of left A-modules for any ring A. More precisely, we construct two cotorsion pairs in Q,AMod from a given cotorsion pair in AMod. This leads to a construction of projective model structures on Q,AMod under the condition that Q has no cycles. We further apply this method to the category Dif(A) of differential left A-modules, viewed as a category of Q-shaped modules for a suitable choice of Q. In this case, the induced cotorsion pairs are shown to be compatible, thereby giving rise to abelian model structures on Dif(A).

We prove two structural results about factorization systems on $\infty$-categories. Firstly, we classify factorization systems on the total spaces of a class of fibrations of $\infty$-categories in terms of factorization systems on the base and the fibers. This will be interpreted as an un/straightening equivalence for $\infty$-categories equipped with a factorization system, using Juran's double $\infty$-categorical framework. Along the way, we develop the general theory of lifting through faithful functors of $n$-uple $\infty$-categories. Secondly, we prove that the forgetful functor from $\infty$-categories equipped with a factorization system to $\infty$-categories is a monadic right adjoint.

We construct an explicit pre-$9$-angulated category which is not $9$-angulated, thereby giving a genuinely higher counterexample to the implication from pre-$(n+2)$-angulated to $(n+2)$-angulated. The underlying additive category is the category of finitely generated projective right modules over the preprojective algebra $\Pi(A_5)$ over $\mathbb F_2$. The construction is obtained by taking an odd power of the twisted complete comparison used by Chen-Liu-Lu-Zhang in their pre-triangulated counterexample and by showing that the resulting pre-$9$-angulation fails the higher mapping-cone axiom.

We prove a conjecture proposed by Dwyer and Hess (2014), namely that the category of symmetric sequences, when equipped with the composition and matrix monoidal structures, forms a 2-monoidal category in the sense of Aguiar and Mahajan (2010). Our approach leverages a detailed analysis of the interplay between these two monoidal structures, offering new insights into their behavior. As a consequence, we are able to show that every operad admits a Day-operad structure.

Strongly bounded objects in a weakly approximable triangulated category are known to relate closely to global dimension and to play a significant role in the uniqueness problem for triangulated enhancements. In this paper, we investigate when the full subcategory of strongly bounded objects admits a bounded $t$-structure. Our main result, under a finiteness condition termed the finite strong finitistic dimension, states that this happens exactly when the subcategory agrees with the full subcategory of bounded objects in the ambient triangulated category. In that case, the bounded $t$-structure is unique up to equivalence; even more, the uniqueness holds unconditionally on the full subcategory of bounded objects.

We examine how the standard proof that left adjoints preserve colimits behaves in the setting of wild categories, a natural setting for synthetic homotopy theory inside homotopy type theory. We show that the proof may fail for adjunctions between wild categories and even produce a wild left adjoint that fails to preserve colimits. Our core contribution, however, is a sufficient condition on the left adjoint for the proof to go through. The condition, which we call 2-coherence, expresses that the naturality structure of the hom-isomorphism commutes with composition of morphisms. We present two useful examples of this condition in action. First, we use it, along with a new version of a known trick for homogeneous types, to show that the suspension functor, as well as a generalization thereof, preserves graph-indexed colimits. Second, we show that every modality, viewed as a functor on coslices of a type universe, is 2-coherent as a left adjoint to the forgetful functor from the subcategory of modal types, thereby proving this subcategory is cocomplete. We have formalized our main results in Agda.