arXiv subjects

Bojana Femić

Publications and source records attributed to Bojana Femić.

18 recordsLinked to original sources

arXiv2026

Bi-representable bi-multicategories

We prove a two-dimensional version of Hermida's equivalence result on representable multicategories and classify its stricter versions. The obtained three-dimensional equivalence of the respective totality categories is of categories enriched over the multicategory $\mathsf{Bicat}$ of Verity. There is a basic enriched equivalence with respect to inherently lax monoidal pseudofunctors, and a one with respect to pseudo-monoidal pseudofunctors, related to the latter via the forgetful enriched functors. We extend Slattery's result on a bi-multicategory structure of the Kleisli bicategory of a 2-multicategory encompassing bi-representability.

math.CT↗

arXiv2025

Para construction for double categories

We introduce Para and coPara double categories for double categories. They rely on a horizontal action $\crta\ot$ of a horizontally monoidal double category $\Mm$ on a double category $\Dd$. We prove a series of properties, most importantly, we characterize monoidality of $\coPara_\Mm(\Dd)$ in the way that it extends monoidality of $\Dd$ as: lax monoidality of the action $\crta\ot$; bistrong commutativity of $\crta\ot$; and purely central premonoidality of $\coPara_\Mm(\Dd)$.

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arXiv2025

Premonoidal and Kleisli double categories

We give a double categorical version of the recently introduced notion of premonoidal bicategories. We introduce a funny product and a funny type of multicategory on double categories granting them a closed funny monoidal structure. We investigate relations between various funny type of structures and premonoidal double categories. We prove that a premonoidal double category $\Dd$ is purely central if and only if its binoidal structure is given by a pseudodouble quasi-functor (a multimap for a Gray type of multicategory) if and only if it admits a monoidal structure. For such $\Dd$ we introduce pure center and show that the monoidal structure on $\Dd$ extends to it. We also discuss one-sided and general centers. Exploiting the companion-lifting properties of vertical structures in a double category into their horizontal counterparts, we prove a series of further results simplifying proofs for the corresponding bicategorical findings. We introduce vertical strengths on vertical double monads and horizontal strengths on horizontal double monads and prove that the former induce the latter. We show that vertical strengths induce actions of the induced horizontally monoidal double category on the corresponding Kleisli double category of the induced horizontal double monad. We prove that there is a 1-1 correspondence between horizontal strengths and extensions of the canonical action of the double category on itself. Finally, we show that for a bistrong vertical double monad the corresponding Kleisli double category is premonoidal.

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arXiv2024

Gray (skew) multicategories: double and Gray-categorical cases

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.

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arXiv2023

Categorical centers and Yetter--Drinfel`d-modules as 2-categorical (bi)lax structures

The bicategorical point of view provides a natural setting for many concepts in the representation theory of monoidal categories. We show that centers of twisted bimodule categories correspond to categories of 2-dimensional natural transformations and modifications between the deloopings of the twisting functors. We also show that dualities lift to centers of twisted bimodule categories. Inspired by the notion of (pre)bimonoidal functors due to McCurdy and Street and by bilax functors of Aguiar and Mahajan, we study 2-dimensional functors which are simultaneously lax and colax with a compatibility condition. Our approach uses a sort of 2-categorical Yang-Baxter operators, but the idea could equally be carried out using a kind of 2-categorical braidings. We show how this concept, which we call bilax functors, generalize many known notions from the theory of Hopf algebras. We propose a 2-category of bilax functors whose 1-cells generalize the notions of Yetter-Drinfel`d modules in ordinary categories, and a type of bimonads and mixed distributive laws in 2-categories. We show that the 2-category of bilax functors from the trivial 2-category is isomorphic to the 2-category of bimonads, and that there is a faithful 2-functor from the latter to the 2-category of mixed distributive laws of Power and Watanabe.

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arXiv2023

The semibicategory of Moore automata

We study the semibicategory $\textsf{Mre}$ of "Moore automata": an arrangement of objects, 1- and 2-cells which is inherently and irredeemably nonunital in dimension one. Between the semibicategory of Moore automata and the better behaved bicategory $\textsf{Mly}$ of "Mealy automata" a plethora of adjunctions insist: the well-known essential equivalence between the two kinds of state machines that model the definitions of $\textsf{Mre}$ and $\textsf{Mly}$ is appreciated at the categorical level, as the equivalence induced between the fixpoints of an adjunction, in fact exhibiting $\textsf{Mre}(A,B)$ as a coreflective subcategory of $\textsf{Mly}(A,B)$; the comodality induced by this adjunction is but the $0$th step of a `level-like' filtration of the bicategory $\textsf{Mre}$ in a countable family of essential bi-localizations $\textsf{s}^n\textsf{Mre}\subseteq\textsf{Mre}$. We outline a way to generate intrinsically meaningful adjunctions of this form. We mechanize some of our main results using the proof assistant Agda.

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arXiv2023

Bifunctor Theorem and strictification tensor product for double categories with lax double functors

We introduce a candidate for the inner hom for $Dbl^{st}_{lx}$, the category of strict double categories and lax double functors, and characterize a lax double functor into it obtaining a lax double quasi-functor. The latter consists of a pair of lax double functors with four 2-cells resembling distributive laws. We extend this characterization to a 2-category isomorphism $q\x\Lax_{hop}(\Aa\times\Bb,\Cc) \iso \Lax_{hop}(\Aa, \llbracket\Bb,\Cc\rrbracket)$. We show that instead of a Gray monoidal product in $Dbl^{st}_{lx}$ we obtain a product that in a sense strictifies lax double quasi-functors. We prove a bifunctor theorem by which certain type of lax double quasi-functors give rise to lax double functors on the Cartesian product, extend it to a 2-functor $q\x\Lax_{hop}^{ns}(\Aa\times\Bb,\Cc)\to\Lax_{hop}(\Aa\times\Bb,\Cc)$ and show how it restricts to a biequivalence. The (un)currying 2-functors are studied. We prove that a lax double functor from the trivial double category is a monad in the codomain double category, and show that our above 2-functor in the form $q\x\Lax_{hop}(*\times *,\Dd)\to\Lax_{hop}(*,\Dd)$ recovers the specification $\Comp(\HH(\Dd)):\Mnd\Mnd(\HH(\Dd))\to\Mnd(\HH(\Dd))$ of the natural transformation $\Comp$ introduced by Street, where $\HH(\Dd)$ is the horizontal 2-category of $\Dd$.

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arXiv2022

Internalization and enrichment via spans and matrices in a tricategory

We introduce categories $\M$ and $§$ internal in the tricategory $\Bicat_3$ of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory $V$. Their horizontal tricategories are the tricategories of matrices and spans in $V$. Both the internal and the enriched constructions are tricategorifications of the corresponding constructions in 1-categories. Following \cite{FGK} we introduce monads and their vertical morphisms in categories internal in tricategories. We prove an equivalent condition for when the internal categories for matrices $\M$ and spans $§$ in a 1-strict tricategory $V$ are equivalent, and deduce that in that case their corresponding categories of (strict) monads and vertical monad morphisms are equivalent, too. We prove that the latter categories are isomorphic to those of categories enriched and discretely internal in $V$, respectively. As a byproduct of our tricategorical constructions we recover some results from \cite{Fem}. Truncating to 1-categories we recover results from \cite{CFP} and \cite{Ehr} on the equivalence of enriched and discretely internal 1-categories.

math.CT↗

arXiv2021

Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories

This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which it is not if one uses the usual notion of a monoidal pseudo double category. Secondly, in \cite{Gabi} the question was raised: which would be an alternative notion to intercategories of Grandis and Paré, so that monoids in Böhm's monoidal category $(Dbl,\ot)$ of strict double categories and strict double functors with a Gray type monoidal product be an example of it? We obtain and prove that precisely the monoidal structure of $(Dbl,\ot)$ resolves the first question. On the other hand, resolving the second question, we upgrade the category $Dbl$ to a tricategory $\DblPs$ and propose %an alternative definition of intercategories as to consider internal categories in this tricategory. %, enabling monoids in $(Dbl,\ot)$ to be examples of this gadget. Apart from monoids in $(Dbl,\ot)$ - more importanlty, weak pseudomonoids in a tricategory containing $(Dbl,\ot)$ as a sub 1-category - most of the examples of intercategories are also examples of this gadget, the ones that escape are those that rely on laxness of the product on the pullback, as duoidal categories. For the latter purpose we define categories internal to tricategories (of the type of $\DblPs$), which simultaneously serves our third motive. Namely, inspired by the tricategory and $(1\times 2)$-category of tensor categories, we prove under mild conditions that categories enriched over certain type of tricategories may be made into categories internal in them. We illustrate this occurrence for tensor categories with respect to the ambient tricategory $2\x\Cat_{wk}$ of 2-categories, pseudofunctors, pseudonatural transformations and modifications.

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arXiv2021

Alternative notion to intercategories: part I. A tricategory of double categories

This is a first of a series of two papers. Our motive is to tackle the question raised in Böhm's "The Gray Monoidal Product of Double Categories" from Applied Categorical Structures: which would be an alternative notion to intercategories of Grandis and Paré, so that monoids in Böhm's monoidal category $Dbl$ of strict double categories and double pseudo functors be an example of it? Before addressing this question we observe that although bicategories embed into pseudo double categories, this embedding is not monoidal, with the usual notion of a monoidal pseudo double category. We then prove that monoidal bicategories embed into the mentioned monoids of Böhm. In order to fit Böhm's monoid into an intercategory-type object, we start by upgrading the category $Dbl$ to a 2-category and end up rather with a tricategory $\DblPs$. We propose an alternative definition of intercategories as internal categories in this tricategory, enabling $Dbl$ to be an example of this gadget. The formal definition of a category internal to the (type of a) tricategory (of) $\DblPs$, as well as another important example of these in the literature, we leave for a subsequent paper.

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arXiv2019

Villamayor-Zelinsky sequence for symmetric finite tensor categories

We prove that if a finite tensor category $\C$ is symmetric, then the monoidal category of one-sided $\C$-bimodule categories is symmetric. Consequently, the Picard group of $\C$ (the subgroup of the Brauer-Picard group introduced by Etingov-Nikshych-Gelaki) is abelian in this case. We then introduce a cohomology over such $\C$. An important piece of tool for this construction is the computation of dual objects for bimodule categories and the fact that for invertible one-sided $\C$-bimodule categories the evaluation functor involved is an equivalence, being the coevaluation functor its quasi-inverse, as we show. Finally, we construct an infinite exact sequence a la Villamayor-Zelinsky for $\C$. It consists of the corresponding cohomology groups evaluated at three types of coefficients which repeat periodically in the sequence.

math.QA↗

arXiv2018

A bicategorical approach to actions of monoidal categories

We characterize in terms of bicategories actions of monoidal categories to representation categories of algebras. For that purpose we introduce cocycles in any 2-category $\K$ and the category of Tambara modules over a monad $B$ in $\K$. We show that in an appropriate setting the above action of categories is given by a 2-cocycle in the Eilenberg-Moore category for the monad $B$. Furthermore, we introduce (co)quasi-bimonads in $\K$ and their respective 2-categories. We show that the categories of Tambara (co)modules over a (co)quasi-bimonad in $\K$ are monoidal, and how the 2-cocycles in the Eilenberg-Moore category corresponding to their actions are related to the Sweedler's and Hausser-Nill 2-cocycles in $\K$. We define (strong) Yetter-Drinfel`d modules in $\K$ as 1-endocells of the 2-category $\Bimnd(\K)$ of bimonads in $\K$, which we introduced in a previous paper. We prove that the monoidal category of Tamabra strong Yetter-Drinfel`d modules in $\K$ acts on the category of relative modules in $\K$. Finally, we show how the above-mentioned results on actions of categories come from pseudofunctors between appropriate bicategories. Our results are 2-categorical generalizations of several results known in the literature.

math.QA↗

arXiv2017

Paired wreaths: towards a 2-categorical atlas of cross products

After we introduced biwreaths and biwreath-like objects in our previous paper, in the present one we define paired wreaths. In a paired wreath there is a monad $B$ and a comonad $F$ over the same 0-cell in a 2-category $\K$, so that $F$ is a left wreath around $B$ and $B$ is a right cowreath around $F$, and moreover, $FB$ is a bimonad in $\K$. The corresponding 1-cell $FB$ in the setting of a biwreath and a biwreath-like object was not necessarilly a bimonad. We obtain a 2-categorical version of the Radford biproduct and Sweedler's crossed (co)product, that are on one hand, both a biwreath and a biwreath-like object, respectively, and on the other hand, they are also paired wreaths. We show that many known crossed (bi)products in the literature are special cases of paired wreaths, including cocycle cross product bialgebras of Bespalov and Drabant in braided monoidal categories. This is a part of a project of constructing a kind of a 2-categorical atlas of all the known crossed (bi)products. We introduce a Hopf datum in $\K$ which contains part of the structure of a paired wreath. We define Yang-Baxter equations and naturality conditions of certain distributive laws in $\K$ and study when Hopf data are paired wreaths. From the notion of a Hopf datum new definitions of Yetter-Drinfel`d modules, (co)module (co)monads, 2-(co)cycles and (co)cycle twisted (co)actions in a 2-categorical setting are suggested.

math.QA↗

arXiv2017

Biwreaths: a self-contained system in a 2-category that encodes different known algebraic constructions and gives rise to new ones

We introduce bimonads in a 2-category $\K$ and define biwreaths as bimonads in the 2-category $\bEM(\K)$ of bimonads, in the analogous fashion as Lack and Street defined wreaths. A biwreath is then a system containing a wreath, a cowreath and their mixed versions, but also a 2-cell $λ$ in $\bEM(\K)$ governing the compatibility of the monad and the comonad structure of the biwreath. We deduce that the monad laws encode 2-(co)cycles and the comonad laws so called 3-(co)cycles, while the 2-cell conditions of the (co)monad structure 2-cells of the biwreath encode (co)actions twisted by these 2- and 3-(co)cycles. The compatibilities of $λ$ deliver concrete expressions of the latter structure 2-cells. We concentrate on the examples of biwreaths in the 2-category induced by a braided monoidal category $\C$ and take for the distributive laws in a biwreath the braidings of the different categories of Yetter-Drinfel'd modules in $\C$. We prove that the before-mentioned properties of a biwreath specified to the latter setting recover on the level of $\C$ different algebraic constructions known in the category ${}_R\M$ of modules over a commutative ring $R$, such as Radford biproduct, Sweedler's crossed product algebra, comodule algebras over a quasi-bialgebra and the Drinfel'd twist. In this way we obtain that the known examples of (mixed) wreaths coming from ${}_R\M$ are not merely examples, rather they are consequences of the structure of a biwreath, and that the form of their structure morphisms originates in the laws inside of a biwreath. Choosing different distributive laws and different 2-cells $λ$ in a biwreath, leads to different and possibly new algebraic constructions.

math.QA↗

arXiv2016

Eilenberg-Watts Theorem for 2-categories and quasi-monoidal structures for module categories over bialgebroid categories

We prove Eilenberg-Watts Theorem for 2-categories of the representation categories $\C\x\Mod$ of finite tensor categories $\C$. For a consequence we obtain that any autoequivalence of $\C\x\Mod$ is given by tensoring with a representative of some class in the Brauer-Picard group $\BrPic(\C)$. We introduce bialgebroid categories over $\C$ and a cohomology over a symmetric bialgebroid category. This cohomology turns out to be a generalization of the one we developed in a previous paper and moreover, an analogous Villamayor-Zelinsky sequence exists in this setting. In this context, for a symmetric bialgebroid category $\A$, we interpret the middle cohomology group appearing in the third level of the latter sequence. We obtain a group of quasi-monoidal structures on the representation category $\A\x\Mod$.

math.QA↗

arXiv2015

Coring categories and Villamayor-Zelinsky sequence for symmetric finite tensor categories

In the preceeding paper we constructed an infinite exact sequence a la Villamayor-Zelinsky for a symmetric finite tensor category. It consists of cohomology groups evaluated at three types of coefficients which repeat periodically. In the present paper we interpret the middle cohomology group in the second level of the sequence. We introduce the notion of coring categories and we obtain that the mentioned middle cohomology group is isomorphic to the group of Azumaya quasi coring categories. This result is a categorical generalization of the classical Crossed Product Theorem, which relates the relative Brauer group and the second Galois cohomology group with respect to a Galois field extension. We construct the colimit over symmetric finite tensor categories of the relative groups of Azumaya quasi coring categories and the full group of Azumaya quasi coring categories over $vec$. We prove that the latter two groups are isomorphic.

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arXiv2013

Transparency condition in the categories of Yetter-Drinfel'd modules over Hopf algebras in braided categories

We study versions of the categories of Yetter-Drinfel'd modules over a Hopf algebra $H$ in a braided monoidal category $\C$. Contrarywise to Bespalov's approach, all our structures live in $\C$. This forces $H$ to be transparent or equivalently to lie in Müger's center $\Z_2(\C)$ of $\C$. We prove that versions of the categories of Yetter-Drinfel'd modules in $\C$ are braided monoidally isomorphic to the categories of (left/right) modules over the Drinfel'd double $D(H)\in\C$ for $H$ finite. We obtain that these categories polarize into two disjoint groups of mutually isomorphic braided monoidal categories. We conclude that if $H\in\Z_2(\C)$, then ${}_{D(H)}\C$ embeds as a subcategory into the braided center category $\Z_1({}_H\C)$ of the category ${}_H\C$ of left $H$-modules in $\C$. For $\C$ braided, rigid and cocomplete and a quasitriangular Hopf algebra $H$ such that $H\in\Z_2(\C)$ we prove that the whole center category of ${}_H\C$ is monoidally isomorphic to the category of left modules over $\Aut({}_H\C)\rtimes H$ - the bosonization of the braided Hopf algebra $\Aut({}_H\C)$ which is the coend in ${}_H\C$. A family of examples of a transparent Hopf algebras is discussed.

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arXiv2013

The Hopf automorphism group and the quantum Brauer group in braided monoidal categories

With the motivation of giving a more precise estimation of the quantum Brauer group of a Hopf algebra $H$ over a field $k$ we construct an exact sequence containing the quantum Brauer group of a Hopf algebra in a certain braided monoidal category. Let $B$ be a Hopf algebra in $\C=_H ^H\YD$, the category of Yetter-Drinfel'd modules over $H$. We consider the quantum Brauer group $\BQ(\C; B)$ of $B$ in $\C$, which is isomorphic to the usual quantum Brauer group $\BQ(k; B\rtimes H)$ of the Radford biproduct Hopf algebra $B\rtimes H$. We show that under certain symmetricity condition on the braiding in $\C$ there is an inner action of the Hopf automorphism group of $B$ on the former. We prove that the subgroup $\BM(\C; B)$ - the Brauer group of module algebras over $B$ in $\C$ - is invariant under this action for a family of Radford biproduct Hopf algebras. The analogous invariance we study for $\BM(k; B\rtimes H)$. We apply our recent results on the latter group and generate a new subgroup of the quantum Brauer group of $B\rtimes H$. In particular, we get new information on the quantum Brauer groups of some known Hopf algebras.

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