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The intent of this paper is to explore Dowker duality from a combinatorial, topological, and categorical perspective. The paper presents three short, new proofs of Dowker duality using various poset fiber lemmas. We introduce modifications of joins and products of simplicial complexes called relational join and relational product complexes. These relational complexes can be constructed whenever there is a relation between simplicial complexes, which includes the context of Dowker duality and covers of simplicial complexes. In this more general setting, we show that the homologies of the simplicial complexes and the relational complexes fit together in a long exact sequence. Similar results are then established for profunctors, which are generalizations of relations to categories. The cograph and graph of profunctors play the role of the relational join and relational product complexes. For a profunctor that arise from adjoint functors between $C$ and $D$, we show that $C$, $D$, the cograph, and the graph all have homotopy equivalent classifying spaces. Lastly, we show that given any profunctor from $D$ to $C$, the homologies of $C$, $D$, the cograph, and the graph form a long exact sequence. We make use of a spectral sequence for Grothendieck fibrations of small categories.

In this article, we study the $(m+2)$-angulations on a Riemann surface, characterized with its boundary components, punctures, and gender. We count the number of arcs in such a surface, and associate a graded quiver with superpotential associated with an $(m2)$-angulation. We show the compatibility between the flip of an $(m+2)$-angulation and the flip in the unpunctured case., Comment: 22 pages

We construct a natural morphism $\rho$ from the nerve $\text{MC}_\bullet(L) = \text{MC}(\Omega_\bullet \widehat{\otimes} L)$ of a pronilpotent curved L${}_\infty$-algebra $L$ to the simplicial subset $\gamma_\bullet(L) = \text{MC}(\Omega_\bullet \widehat{\otimes} L,s_\bullet)$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $\gamma_\bullet(L) \hookrightarrow \text{MC}_\bullet(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_\infty$-algebras to curved L${}_\infty$-algebras. The morphism $\rho$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $\rho^\square$ of $\rho$ to identify $\rho$ with higher holonomy for semiabelian curved \Linf-algebras., Comment: 17 pages

This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left $\mathfrak R$-contramodules is equivalent to the derived category of the exact category of flat left $\mathfrak R$-contramodules. In other words, a complex of flat $\mathfrak R$-contramodules is contraacyclic in the sense of Becker if and only if it is an acyclic complex with flat $\mathfrak R$-contramodules of cocycles., Comment: LaTeX 2e, 20 pages. arXiv admin note: text overlap with arXiv:2306.02734

We elaborate an algebraic framework for describing internal topological symmetries of gapped boundaries of (2+1)D topological orders. We present a categorical obstruction to the coherence of bulk group symmetry and boundary symmetries in terms of liftings of categorical actions on the bulk theory to a certain 2-group of boundary symmetries., Comment: 18 pages, 11 figures. Comments welcome

We introduce necklicial nerve functors from enriched categories to simplicial sets, which include Cordier's homotopy coherent, Lurie's differential graded and Le Grignou's cubical nerves. It is shown that every necklicial nerve can be lifted to the templicial objects of arXiv:2302.02484v2. Building on the work of Dugger and Spivak, we give sufficient conditions under which the left-adjoint of a necklicial nerve can be described more explicitly. As an application, we obtain novel and simple expressions for the left-adjoints of the dg-nerve and cubical nerve., Comment: 49 pages, no figures

These expanded lecture notes are based on a tutorial on categorical proof theory presented at the summer school associated with the conference "Topology, Algebra, and Categories in Logic 2021-2022." The chapter delves into various applications of categorical methods in proof theory. It is designed to be accessible, with no prior familiarity with category theory required. The necessary categorical background is introduced gradually, with a focus on the philosophical and informal aspects of proof. The only prerequisites are a basic understanding of logic, computability theory, topology, and ordered structures., Comment: 130 pages

Garret Birkhoff observed that any binary relation between two sets determines a Galois connection between the powersets, or equivalently, closure operators on the powersets, or equivalently, complete lattices of subsets that are dually isomorphic. Referring to the duality of, say, points and lines in projective geometry, he named the binary relations as polarities. Researchers since then have used polarities (also known as formal contexts) as a convenient technical way to build complete lattices from ``found'' data. And so, various proposals for suitable morphisms between polarities have tended to have a particular application in mind. In this work, we develop the structure of a category of polarities and compatible relations, adopting Birkhoff's original simple idea that the structure of a polarity is its the Galois connection. Hence, morphisms must be relations that, in a reasonable sense, preserve Galois connections. In particular, the dual equivalence of the category to the category of complete meet semilattices, completeness of the category, characterization of epimorphisms and monomorphisms, an epi/mono factorization system, as well as the star-autonomous structure of the category, all arise by extending Birkhoff's original observation to morphisms.

Over the topos of sets, the notion of Lawvere theory is infinite countably-sorted algebraic but not one-sorted algebraic. Shifting viewpoint over the object-classifier topos, a finite algebraic presentation of Lawvere theories is considered.

We introduce a novel concept of action for unitary magmas, facilitating the classification of various split extensions within this algebraic structure. Our method expands upon the recent study of split extensions and semidirect products of unitary magmas conducted by Gran, Janelidze, and Sobral. Building on their research, we explore split extensions in which the middle object does not necessarily maintain a bijective correspondence with the Cartesian product of its end objects. Although this phenomenon is not observed in groups or any associative semiabelian variety of universal algebra, it shares similarities with instances found in monoids through weakly Schreier extensions and certain exotic non-associative algebras, such as semi-left-loops. Our work seeks to contribute to the comprehension of split extensions in unitary magmas and may offer valuable insights for potential abstractions of categorical properties in more general contexts., Comment: Unitary magma, Split extensions, Retraction points, Non-associative algebra, Semidirect products, Weakly Schreier extensions

With a complete residuated lattice $L$ as the truth value table, we extend the definition of sobriety of classical convex spaces to the framework of $L$-convex spaces. We provide a specific construction for the sobrification of an $L$-convex space, demonstrating that the full subcategory of sober $L$-convex spaces is reflective in the category of $L$-convex spaces with convexity-preserving mappings. Additionally, we introduce the concept of Scott $L$-convex structures on $L$-ordered sets. As an application of this type of sobriety, we obtain a characterization for the $L$-join-semilattice completion of an $L$-ordered set: an $L$-ordered set $Q$ is an $L$-join-semilattice completion of an $L$-ordered set $P$ if and only if the Scott $L$-convex space $(Q, \sigma^{\ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P, \sigma^{\ast}(P))$.

We prove that there exist Hopf algebras with surjective, non-bijective antipode which admit no non-trivial morphisms from Hopf algebras with bijective antipode; in particular, they are not quotients of such. This answers a question left open in prior work, and contrasts with the dual setup whereby a Hopf algebra has injective antipode precisely when it embeds into one with bijective antipode. The examples rely on the broader phenomenon of realizing pre-specified subspace lattices as comodule lattices: for a finite-dimensional vector space $V$ and a sequence $(\mathcal{L}_r)_r$ of successively finer lattices of subspaces thereof, assuming the minimal subquotients of the supremum $\bigvee_r \mathcal{L}_r$ are all at least 2-dimensional, there is a Hopf algebra equipping $V$ with a comodule structure in such a fashion that the lattice of comodules of the $r^{th}$ dual comodule $V^{r*}$ is precisely the given $\mathcal{L}_r$., Comment: 13 pages + references

In this paper we prove the existence of an algebraic model for quasi-coherent sheaves on certain non-connective geometric stacks arising in stable homotopy theory and spectral algebraic geometry using the machinery of adapted homology theories., Comment: Based on author's thesis; Submitted to Homology, Homotopy, and Applications

$\DeclareMathOperator{\G}{\mathbb{G}}\DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Corr}{Corr}$Let $\G$ be a locally compact quantum group and $(M, \alpha)$ a $\G$-$W^*$-algebra. The object of study of this paper is the $W^*$-category $\Rep^{\G}(M)$ of normal, unital $\G$-representations of $M$ on Hilbert spaces endowed with a unitary $\G$-representation. This category has a right action of the category $\Rep(\G)= \Rep^{\G}(\mathbb{C})$ for which it becomes a right $\Rep(\G)$-module $W^*$-category. Given another $\G$-$W^*$-algebra $(N, \beta)$, we denote the category of normal $*$-functors $\Rep^{\G}(N)\to \Rep^{\G}(M)$ compatible with the $\Rep(\G)$-module structure by $\operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ and we denote the category of $\G$-$M$-$N$-correspondences by $\operatorname{Corr}^{\G}(M,N)$. We prove that there are canonical functors $P: \Corr^{\G}(M,N)\to \operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ and $Q: \operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))\to \operatorname{Corr}^{\G}(M,N)$ such that $Q \circ P\cong \operatorname{id}.$ We use these functors to show that the $\G$-dynamical von Neumann algebras $(M, \alpha)$ and $(N, \beta)$ are equivariantly Morita equivalent if and only if $\Rep^{\G}(N)$ and $\Rep^{\G}(M)$ are equivalent as $\Rep(\G)$-module-$W^*$-categories. Specializing to the case where $\G$ is a compact quantum group, we prove that moreover $P\circ Q \cong \operatorname{id}$, so that the categories $\Corr^{\G}(M,N)$ and $\operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ are equivalent. This is an equivariant version of the Eilenberg-Watts theorem for actions of compact quantum groups on von Neumann algebras., Comment: 20 pages, comments are welcome!

The coefficient categories of six functor formalisms are often locally rigid, and when this is the case, the exceptional pushforward and pullback adjunctions may be defined formally. In this short note it is shown that for f a proper map resp. an open embedding the well known formulas f_! = f_* resp. f_! = f_# may likewise be deduced formally., Comment: 9 pages

We investigate approach spaces generated by probabilistic metric spaces with respect to a continuous t-norm $*$ on the unit interval $[0,1]$. Let $k^*$ be the supremum of the idempotent elements of $*$ in $[0,1)$. It is shown that if $k^*=1$ (resp. $k^*<1$), then an approach space is probabilistic metrizable with respect to $*$ if and only if it is probabilistic metrizable with respect to the minimum (resp. product) t-norm., Comment: 15 pages

Intensional computation derives concrete outputs from abstract function definitions; extensional computation defines functions through explicit input-output pairs. In formal semantics: intensional computation interprets expressions as context-dependent functions; extensional computation evaluates expressions based on their denotations in an otherwise fixed context. This paper reformulates typed extensional and intensional models of formal semantics within a category-theoretic framework and demonstrates their natural representation therein. We construct $\textbf{ModInt}$, the category of intensional models, building on the categories $\textbf{Set}$ of sets, $\textbf{Rel}$ of relations, and $\textbf{Kr}$ and $\textbf{Kr}_\textbf{b}$ of Kripke frames with monotone maps and bounded morphisms, respectively. We prove that trivial intensional models are equivalent to extensional models, providing a unified categorical representation of intensionality and extensionality in formal semantics. This approach reinterprets the relationship between intensions and extensions in a categorical framework and offers a modular, order-independent method for processing intensions and recovering extensions; contextualizing the relationship between content and reference in category-theoretic terms. We discuss implications for natural language semantics and propose future directions for contextual integration and exploring $\textbf{ModInt}$'s algebraic properties., Comment: 28 pages, 8 figures, minor edits

Compact metric spaces form an important class of metric spaces, but the category that they define lacks many important properties such as completeness and cocompleteness. In recent studies of "metric domain theory" and Stone-type dualities, the more general notion of a (separated) metric compact Hausdorff space emerged as a metric counterpart of Nachbin's compact ordered spaces. Roughly speaking, a metric compact Hausdorff space is a metric space equipped with a \emph{compatible} compact Hausdorff topology (which does not need to be the induced topology). These spaces maintain many important features of compact metric spaces, and, notably, the resulting category is much better behaved. Moreover, one can use inspiration from the theory of Nachbin's compact ordered spaces to solve problems for metric structures. In this paper we continue this line of research: in the category of separated metric compact Hausdorff spaces we characterise the regular monomorphisms as the embeddings and the epimorphisms as the surjective morphisms. Moreover, we show that epimorphisms out of an object $X$ can be encoded internally on $X$ by their kernel metrics, which are characterised as the continuous metrics below the metric on $X$; this gives a convenient way to represent quotient objects. Finally, as the main result, we prove that its dual category has an algebraic flavour: it is Barr-exact. While we show that it cannot be a variety of finitary algebras, it remains open whether it is an infinitary variety.

Inspired by the study of vertex operator algebra extensions, we answer the question of when the category of local modules over a commutative exact algebra in a braided finite tensor category is a (non-semisimple) modular tensor category. Along the way we provide sufficient conditions for the category of local modules to be rigid, pivotal and ribbon. We also discuss two ways to construct such commutative exact algebras. The first is the class of simple current algebras and the second is using right adjoints of central tensor functors. Furthermore, we discuss Witt equivalence and its relation with extensions of VOAs., Comment: v1: 48 pages. Comments welcome

We prove a connectivity bound for maps of $\infty$-operads of the form $\mathbb{A}_{k_1} \otimes \cdots \otimes \mathbb{A}_{k_n} \to \mathbb{E}_n$, and as a consequence, give an inductive way to construct $\mathbb{E}_n$-algebras in $m$-categories. The result follows from a version of Eckmann-Hilton argument that takes into account both connectivity and arity of $\infty$-operads. Along the way, we prove a technical Blakers-Massey type statement for algebras of coherent $\infty$-operads., Comment: 21 pages, 1 table, comments are welcome

It is shown that a quiver is left noetherian if and only if the category of quiver representations in any locally noetherian abelian category is again locally noetherian. Here, locally noetherian means that any object is the directed union of its noetherian subobjects. For a quiver to be left noetherian means that the left ideals of paths starting at any fixed vertex satisfy the ascending chain condition. The proof generalises to representations of any small category that admits a Gr\"obner enrichment., Comment: 10 pages

In this paper we describe the homotopy category of the $A_\infty$categories. To do that we introduce the notion of semi-free $A_\infty$category, which plays the role of standard cofibration. Moreover, we define the non unital $A_\infty$ (resp. DG)categories with cofibrant morphisms and we prove that any non unital $A_\infty$ (resp. DG)category has a resolution of this kind., Comment: 49 pages, 5 figures

We give a new construction of the model structure on the category of simplicial sets for homotopy $n$-types, originally due to Elvira-Donazar and Hernandez-Paricio, using a right transfer along the coskeleton functor. We observe that an analogous model structure can be constructed on the category of cubical sets, and use it to equip the category of (simple) graphs with a fibration category structure whose weak equivalences are discrete $n$-equivalences., Comment: 25 pages; comments welcome

We use the Symmetry Topological Field Theory (SymTFT) to study and classify gapped phases in (2+1)d for a class of categorical symmetries, referred to as being of bosonic type. The SymTFTs for these symmetries are given by twisted and untwisted (3+1)d Dijkgraaf-Witten (DW) theories for finite groups G. A finite set of boundary conditions (BCs) of these DW theories is well-known: these simply involve imposing Dirichlet and Neumann conditions on the (3+1)d gauge fields. We refer to these as minimal BCs. The key new observation here is that for each DW theory, there exists an infinite number of other BCs, that we call non-minimal BCs. These non-minimal BCs are all obtained by a 'theta construction', which involves stacking the Dirichlet BC with 3d TFTs having G 0-form symmetry, and gauging the diagonal G symmetry. On the one hand, using the non-minimal BCs as symmetry BCs gives rise to an infinite number of non-invertible symmetries having the same SymTFT, while on the other hand, using the non-minimal BCs as physical BCs in the sandwich construction gives rise to an infinite number of (2+1)d gapped phases for each such non-invertible symmetry. Our analysis is thoroughly exemplified for G = $\mathbb{Z_2}$ and more generally any finite abelian group, for which the resulting non-invertible symmetries and their gapped phases already reveal an immensely rich structure., Comment: 83 pages

Locality is implemented in an arbitrary category using Grothendieck topologies. We explore how different Grothendieck topologies on one category can be related, and, more general, how functors between categories can preserve them. As applications of locality, we review geometric objects such as sheaves, groupoids, functors, bibundles, and anafunctors internal to an arbitrary Grothendieck site. We give definitions such that all these objects are invariant under equivalences of Grothendieck topologies and certain functors between sites. As examples of sites, we look at categories of smooth manifolds, diffeological spaces, topological spaces, and sheaves, and we study properties of various functors between those.

We show that the notion of $(\infty,n)$-limit defined using the enriched approach and the one defined using the internal approach coincide. We also give explicit constructions of various double $(\infty,n-1)$-categories implementing various join constructions, slice constructions and cone constructions, and study their properties. We further prove that key examples of $(\infty,n)$-categories are (co)complete., Comment: 45 pages; comments welcome

We compute (exact and asymptotic) formulas for the growth rate of the number of indecomposable summands in the tensor powers of representations of finite groups, over a field of arbitrary characteristic. In characteristic zero, we obtain in addition a general exact formula for the growth rate and give a complete solution to the growth problems in terms of the character table. We also provide code used to compute our formulas., Comment: 18 pages, many figures, comments welcome

In this paper, by means of upper approximation operators in rough set theory, we study representations for sL-domains and its special subclasses. We introduce the concepts of sL-approximation spaces, L-approximation spaces and bc-approximation spaces, which are special types of CF-approximation spaces. We prove that the collection of CF-closed sets in an sL-approximation space (resp., an L-approximation space, a bc-approximation space) ordered by set-theoretic inclusion is an sL-domain (resp., an L-domain, a bc-domain); conversely, every sL-domain (resp., L-domain, bc-domain) is order-isomorphic to the collection of CF-closed sets of an sL-approximation space (resp., an L-approximation space, a bc-approximation space). Consequently, we establish an equivalence between the category of sL-domains (resp., L-domains) with Scott continuous mappings and that of sL-approximation spaces (resp., L-approximation spaces) with CF-approximable relations.

In this paper, concepts of (topological) FS-approximation spaces are introduced. Representations of FS-domains and BF-domains via (topological) FS-approximation spaces are considered. It is proved that the collection of CF-closed sets in an FS-approximation space (resp., a topological FS-approximation space) endowed with the set-inclusion order is an FS-domain (resp., a BF-domain) and that every FS-domain (resp., BF-domain) is order isomorphic to the collection of CF-closed sets of some FS-approximation space (resp., topological FS-approximation space) endowed with the set-inclusion order. The concept of topological BF-approximation spaces is introduced and a skillful method without using CF-approximable relations to represent BF-domains is given. It is also proved that the category of FS-domains (resp., BF-domains) with Scott continuous maps as morphisms is equivalent to that of FS-approximation spaces (resp., topological FS-approximation spaces) with CF-approximable relations as morphisms.

In this work, we continue the investigation of certain enrichments of dual algebraic structures in monoidal double categories, that was initiated in [Vas19]. First, we re-visit monads and comonads in double categories and establish a tensored and cotensored enrichment of the former in the latter, under general conditions. These include monoidal closedness and local presentability of the double category, notions that are proposed as tools required for our main results, but are of interest in their own right. The natural next step involves categories of the newly introduced modules for monads and comodules for comonads in double categories. After we study their main categorical properties, we establish a tensored and cotensored enrichment of modules in comodules, as well as an enriched fibration structure that involves (co)modules over (co)monads in double categories. Applying this abstract double categorical framework to the setting of V-matrices produces an enrichment of the category of V-enriched modules (fibred over V-categories) in V-enriched comodules (opfibred over V-cocategories), which is the many-object generalization of the respective result for modules (over algebras) and comodules (over coalgebras) in monoidal categories., Comment: 59 pages

We develop the theory of groupoid graded semisimple rings. Our rings are neither unital nor one-sided artinian. Instead, they exhibit a strong version of having local units and being locally artinian, and we call them $\Gamma_0$-artinian. One of our main results is a groupoid graded version of the Wedderburn-Artin Theorem, where we characterize groupoid graded semisimple rings as direct sums of graded simple $\Gamma_0$-artinian rings and we exhibit the structure of this latter class of rings. In this direction, we also prove a groupoid graded version of Jacobson-Chevalley density theorem. We need to define and study properties of groupoid gradings on matrix rings (possibly of infinite size) over groupoid graded rings, and specially over groupoid graded division rings. Because of that, we study groupoid graded division rings and their graded modules. We consider a natural notion of freeness for groupoid graded modules that, when specialized to group graded rings, gives the usual one, and show that for a groupoid graded division ring all graded modules are free (in this sense). Contrary to the group graded case, there are groupoid graded rings for which all graded modules are free according to our definition, but they are not graded division rings. We exhibit an easy example of this kind of rings and characterize such class among groupoid graded semisimple rings. We also relate groupoid graded semisimple rings with the notion of semisimple category defined by B. Mitchell. For that, we show the link between functors from a preadditive category to abelian groups and graded modules over the groupoid graded ring associated to this category, generalizing a result of P. Gabriel. We characterize simple artinian categories and categories for which every functor from them to abelian groups is free in the sense of B. Mitchell., Comment: 87 pages

We use the inclusion of injectives into the canonical heart as a replacement for tilting objects in computations of the characteristic morphism. Then we apply this construction to proofs of non-liftability of candidate non-Fourier-Mukai functors, i.e.\ functors that do not admit an $\mathcal{A}_\infty$/$\mathrm{dg}$-lift. This approach allows explicit computation of the obstruction against an $\mathcal{A}_\infty$-lift. We in particular observe that this computation gives for smooth degree $d>2$ hypersurfaces an abundance of non-Fourier-Mukai functors., Comment: 24 pages, comments welccome

We present a thorough introduction to the tools of category theory required for formulating gauge theories based on 2-connections. We provide a detailed construction of the categorical generalization of BF theory, dubbed BFCG, also known as 2BF. Similar to BF gravity, it is known that BFCG can be deformed to give general relativity. Here, we obtain an alternative relation between BFCG and gravity, which consists of coupling general relativity and BFCG by means of the volume form constructed out of the BFCG connections. The resulting theory, closely related to unimodular gravity, is a generalization of BF sequestered gravity not only in the sense that it adds new fields but also in that it allows for new choices for the volume form that is coupled to gravity. Furthermore, we show that BF sequestered gravity in the abelian case is recovered for a specific choice of the 2-group.

In this paper we study the indecomposable module categories over $\mathcal{C}(\mathfrak{sl}_N, k)$, the category of integrable level-$k$ respresentations of affine Kac-Moody $\mathfrak{sl}_N$. Our first main result classifies these module categories in the case of generic $k$, i.e. $k$ is sufficiently large relative to $N$. As $\mathcal{C}(\mathfrak{sl}_N, k)$ is a braided tensor category, there is a relative tensor product structure on its category of module categories. In the generic setting we obtain a formula for the relative tensor product rules between the indecomposable module categories. Our second main result classifies the indecomposable module categories over $\mathcal{C}(\mathfrak{sl}_N, k)$ for $N\leq 7$, with no restrictions on $k$. In this non-generic setting, exceptional module categories are obtained. This work relies heavily on previous results by the two authors. In previous literature, module category classification results were known only for $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$., Comment: 37 pages

Given a premodular category $\mathcal{C}$, we show that its $R$-symbol can be recovered from its $T$-matrice, fusion coefficients and some 2nd generalized Frobenius-Schur indicators. In particular, if $\mathcal{C}$ is modular, its $R$-symbols for a certain gauge choice are completely determined by its modular data., Comment: 9 pages latex

Monoidal categories with additional structure such as a braiding or some form of duality abound in quantum topology. They often appear in tandem with Frobenius algebras inside them. Motivations for this range from the theory of module categories to the construction of correlators in conformal field theory. We generalize the Baez-Dolan microcosm principle to consistently describe all these types of algebras by extending it to cyclic and modular algebras in the sense of Getzler-Kapranov. Our main result links the microcosm principle for cyclic algebras to the one for modular algebras via Costello's modular envelope. The result can be understood as a local-to-global construction for various flavors of Frobenius algebras that substantially generalizes and unifies the available, and often intrinsically semisimple methods using for example triangulations, state-sum constructions or skein theory. Several applications of the main result in conformal field theory are presented: We classify consistent systems of correlators for open conformal field theories and show that the genus zero correlators for logarithmic conformal field theories constructed by Fuchs-Schweigert can be uniquely extended to handlebodies. This establishes a very general correspondence between full genus zero conformal field theory in dimension two and skein theory in dimension three., Comment: 29 pages, 4 figure, some diagrams

We establish a one-to-one correspondence between the singularity categories of rational double points and the simply-laced Dynkin graphs in arbitrary characteristic. This correspondence is well-known in characteristic zero since the rational double points are quotient singularities in characteristic zero whereas not necessarily in positive characteristic. Considering some rational double points are not taut in characteristic two, three or five, we can see there exist two rational double points which are not analytically isomorphic but whose singularity categories are triangulated equivalent. As an application, we construct a counter-example in positive characteristic of a theorem of Hua and Keller: the dg singularity category of a hypersurface singularity determines its Tyurina algebra.

We show that if $\alpha$ is a regular cardinal, $\mathcal{D}$ is an $\alpha$-compactly generated triangulated category, in the sense of Neeman \cite{N}, and $\tau$ is a t-structure in $\mathcal{D}$ generated by a set of $\alpha$-compact objects, then the heart of $\tau$ is a locally $\alpha$-presentable (not necessarily Ab5) abelian category. As a consequence, in a well-generated triangulated category any t-structure generated by a set of objects has a heart with a set of generators., Comment: 22 pages

Let $\Lambda$ be a finite dimensional algebra. Let $\mathcal C$ be a functorially finite exact subcategory of $\Lambda$-mod with enough projective and injective objects and $\mathcal S (\mathcal C)$ be its monomorphism category. It turns out that the category $\mathcal S (\mathcal C)$ has almost split sequences. We show an explicit formula for the Auslander-Reiten translation in $\mathcal S (\mathcal C)$. Furthermore, if $\mathcal C$ is a stably $d$-Calabi-Yau Frobenius category, we calculate objects under powers of Auslander-Reiten translation in the triangulated category $\overline{\mathcal S(\mathcal C)}$., Comment: 26 pages

We construct in a unifying way skew-multicategories and multicategories of double and Gray-categories that we call Gray (skew) multicategories. We study their different versions depending on the types of functors and higher transforms. We construct Gray type products by generators and relations and prove that Gray skew-multicategories are closed and representable on one side, and that the Gray multicaticategories taken with the strict type of functors are representable. We conclude that the categories of double and Gray-categories with strict functors underlying Gray (skew) multicategories are skew monoidal, respectively monoidal, depending on the type of the inner-hom and product considered. The described Gray (skew) multicategories we see as prototypes of general Gray (skew) multicategories, which correspond to (higher) categories of higher dimensional internal and enriched categories.

Let $p: \mathcal{E} \to \mathcal{S}$ be a pre-cohesive geometric morphism. We show that the least subtopos of $\mathcal{E}$ containing both the subcategories $p^*: \mathcal{S} \to \mathcal{E}$ and $p^!: \mathcal{S} \to \mathcal{E}$ exists, and that it coincides with the least subtopos containing $p^*2$, where 2 denotes the subobject classifier of $\mathcal{S}$.

Let $\mathsf{Q}$ be a commutative and unital quantale. By a $\mathsf{Q}$-map we mean a left adjoint in the quantaloid of sets and $\mathsf{Q}$-relations, and by a partial $\mathsf{Q}$-map we refer to a Kleisli morphism with respect to the maybe monad on the category $\mathsf{Q}\text{-}\mathbf{Map}$ of sets and $\mathsf{Q}$-maps. It is shown that every $\mathsf{Q}$-map is symmetric if and only if $\mathsf{Q}$ is weakly lean, and that every $\mathsf{Q}$-map is exactly a map in $\mathbf{Set}$ if and only $\mathsf{Q}$ is lean. Moreover, assuming the axiom of choice, it is shown that the category of sets and partial $\mathsf{Q}$-maps is monadic over $\mathsf{Q}\text{-}\mathbf{Map}$., Comment: 18 pages

Let $\mathfrak{C}$ be a symmetric tensor category and let $A$ be an Azumaya algebra in $\mathfrak{C}$. Assuming a certain invariant $\eta(A) \in \mathrm{Pic}(\mathfrak{C})[2]$ vanishes, and fixing a certain choice of signs, we show that there is a universal tensor functor $\Phi \colon \mathfrak{C} \to \mathfrak{D}$ for which $\Phi(A)$ splits. We apply this when $\mathfrak{C}=\underline{\mathrm{Rep}}(\mathbf{Sp}_t(\mathbf{F}_q))$ is the interpolation category of finite symplectic groups and $A$ is a certain twisted group algbera in $\mathfrak{C}$, and we show that the splitting category $\mathfrak{D}$ is S. Kriz's interpolation category of the oscillator representation. This construction has a number advantages over previous ones; e.g., it works in non-semisimple cases. It also brings some conceptual clarity to the situation: the existence of Kriz's category is tied to the non-triviality of the Brauer group of $\mathfrak{C}$., Comment: 31 pages

In arXiv:1712.00555, H. Heine shows that given a symmetric monoidal $\infty$-category $\mathcal{V}$ and a weakly $\mathcal{V}$-enriched monad $T$ over an $\infty$-category $\mathcal{C}$, then there is an induced action of $\mathcal{V}$ on $LMod_T(\mathcal{C})$. Moreover, properties like tensoring or enrichment can be transferred from the action on $\mathcal{C}$ to that on $LMod_T(\mathcal{C})$. We see that the action of an internal operad $O \in Alg(sSeq(\mathcal{C}))$ can be interpreted as the action of a monad $T_O$, such that $Alg_O(\mathcal{C})\cong LMod_{T_O}(\mathcal{C})$. We can then prove that, under a presentability assumption, if the category $\mathcal{C}$ admits cotensors with respect to the action of $\mathcal{V}$, then so does $Alg_O(\mathcal{C})\cong LMod_{T_O}(\mathcal{C})$. This is used to show that the coproduct-idempotent algebras are fixed by the induced tensoring action. We apply this to the stable motivic homotopy category and prove that the tensor of any motivic sphere with rational motivic cohomology is equivalent to the latter., Comment: 32 pages, 1 figure

In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary $\infty$-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a "normal closure" monad in a way which generalizes the standard loops-suspension monad. We generalize a result of Prasma by showing that monoidal functors of $\infty$-topoi preserve normal maps or, equivalently, that monoidal functors of $\infty$-topoi preserve the property of "being a fiber" for morphisms between connected objects. We also formulate Noether's Isomorphism Theorems in this setting, prove the first of them, and provide counterexamples to the other two. Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in $\infty$-topoi in the "usual way." One nice result of this labor is the formulation and proof of an Orbit-Stabilizer Theorem for group actions in $\infty$-topoi., Comment: minor changes, added some references, fixed some typos

Since 1883, Picard-Vessiot theory had been developed as the Galois theory of differential field extensions associated to linear differential equations. Inspired by categorical Galois theory of Janelidze, and by using novel methods of precategorical descent applied to algebraic-geometric situations, we develop a Galois theory that applies to morphisms of differential schemes, and vastly generalises the linear Picard-Vessiot theory, as well as the strongly normal theory of Kolchin.

We establish a calculus of differences for taut endofunctors of the category of sets, analogous to the classical calculus of finite differences for real valued functions. We study how the difference operator interacts with limits and colimits as categorical versions of the usual product and sum rules. The first main result is a lax chain rule which has no counterpart for mere functions. We also show that many important classes of functors (polynomials, analytic functors, reduced powers, ...) are taut, and calculate explicit formulas for their differences. Covariant Dirichlet series are introduced and studied. The second main result is a Newton summation formula expressed as an adjoint to the difference operator., Comment: 80 pages

We introduce the notion of Grothendieck heaps for unpointed Waldhausen categories and unpointed stable $\infty$-categories. This allows an extension of the studies of $\mathrm{K}_0$ to the homotopy category of unpointed topological spaces., Comment: 11 pages, Comments welcome

Over a field of characteristic $p>0$, the higher Verlinde categories $\mathrm{Ver}_{p^n}$ are obtained by taking the abelian envelope of a quotient of the category of tilting modules for the algebraic group $\mathrm{SL}_2$. These symmetric tensor categories have been introduced in arXiv:2003.10499 & arXiv:2003.10105, and their properties have been extensively studied in the former reference. In arXiv:2105.07724, the above construction for $\mathrm{SL}_2$ has been generalized to Lusztig's quantum group for $\mathfrak{sl}_2$ and root of unity $\zeta$, which produces the mixed higher Verlinde categories $\mathrm{Ver}^{\zeta}_{p^{(n)}}$. Inspired by the results of arXiv:2003.10499, we study the properties of these braided tensor categories in detail. In particular, we establish a Steinberg tensor product formula for the simple objects of $\mathrm{Ver}^{\zeta}_{p^{(n)}}$, construct a braided embedding $\mathrm{Ver}_{p^n}\hookrightarrow \mathrm{Ver}^{\zeta}_{p^{(n+1)}}$, compute the symmetric center of $\mathrm{Ver}^{\zeta}_{p^{(n)}}$, and identify its Grothendieck ring., Comment: Comments welcome!

We characterize the families of bialgebras or Hopf algebras over fields for which the product in the corresponding category is finite-dimensional, answering a question of M. Lorenz: if the ground field is infinite then bialgebra or Hopf products are finite-dimensional precisely when the factors are, with at most one of dimension $>1$; over finite fields the necessary and sufficient condition is instead that factors be finite-dimensional with at most finitely many of dimension $>1$; finally, these statements hold for coalgebras as well, provided the family is finite. We also characterize (a) finite field extensions as precisely those whose underlying scalar extension functor preserves coalgebra, or bialgebra, or Hopf algebra products (correcting an error in the literature); (b) algebraic field extensions as those along which finite coalgebra (bialgebra, Hopf algebra) products are preserved; and (c) again algebraic field extensions as precisely those which intertwine cofree coalgebras on vector spaces, or cofree bialgebras (Hopf algebras) on algebras., Comment: v2 adds Theorem 2.13 and makes attendant changes; 24 pages + references