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Josep Elgueta

Publications and source records attributed to Josep Elgueta.

14 recordsLinked to original sources
arXiv2026

On the (algebraic) notion of 2-ring

By a 2-ring we mean a groupoid with a structure analogous to that of a ring, up to coherent isomorphisms. Two different notions of 2-ring appear in the literature: the notion of {\em Ann-category}, due to Quang, and the notion of {\em categorical ring}, due to Jibladze and Pirashvili. The underlying data are the same in both cases, but the required axioms differ. In this note, we clarify the relationship between these notions by explaining why an additional axiom must be imposed for the two notions to be equivalent. Essential to this analysis is an equivalent description of a symmetric monoidal category.

math.CT↗
arXiv2025

An alternative description of symmetric monoidal categories, and symmetric 2-groups

An equivalent description of a symmetric monoidal category is introduced in which, instead of separate associator and commutator isomorphisms satisfying the usual coherence axioms, we simply have associo-commutator isomorphisms satisfying their own coherence laws. In particular, this yields an alternative description of a symmetric 2-group and leads to a cohomological classification of these objects in terms of Eilenberg-MacLane cubical cohomology for abelian groups.

math.CT↗
arXiv2022

Normal submonoids and congruences on a monoid

A notion of {\em normal submonoid} of a monoid $M$ is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf{NorSub}(M)$ of normal submonoids of $M$ is a complete lattice. Joins are explicitly described, and the lattice is computed for the finite full transformation monoids $T_n$, $n\geq 1$. It is also shown that $\mathsf{NorSub}(M)$ is modular for a specific family of commutative monoids, including all Krull monoids, and that, as a join semilattice, embeds isomorphically onto a join subsemilattice of the lattice $\mathsf{Cong}(M)$ of congruences on $M$. This leads to a new strategy for computing $\mathsf{Cong}(M)$ consisting of computing $\mathsf{NorSub}(M)$, and the lattices of the so called unital congruences on the quotients of $M$ modulo its normal submonoids. This provides a new perspective on Malcev computation of the congruences on $T_n$.

math.GR↗
arXiv2020

Endomorphisms of the symmetric 2-rig of finite sets

Let $\widehat{\mathbb{F}\mathbb{S}et}$ be the groupoid of finite sets and bijections between them equipped with the canonical symmetric rig category structure given by the disjoint union and the cartesian product of finite sets. We prove that the category (in fact, groupoid) of endomorphisms of $\widehat{\mathbb{F}\mathbb{S}et}$ is equivalent to the terminal category, thus providing some evidence that $\widehat{\mathbb{F}\mathbb{S}et}$ is the right categorical analog of the commutative rig $\mathbb{N}$ of nonnegative integers. This is shown using a particular semistrict skeletal version of $\widehat{\mathbb{F}\mathbb{S}et}$ for which the endomorphisms can be described very explicitly.

math.CT↗
arXiv2020

The groupoid of finite sets is biinitial in the 2-category of rig categories

The groupoid of finite sets has a "canonical" structure of a symmetric 2-rig with the sum and product respectively given by the coproduct and product of sets. This 2-rig $\widehat{\mathbb{F}\mathbb{S} et}$ is just one of the many non-equivalent categorifications of the commutative rig $\mathbb{N}$ of natural numbers, together with the rig $\mathbb{N}$ itself viewed as a discrete rig category, the whole category of finite sets, the category of finite dimensional vector spaces over a field $k$, etc. In this paper it is shown that $\widehat{\mathbb{F}\mathbb{S} et}$ is the right categorification of $\mathbb{N}$ in the sense that it is biinitial in the 2-category of rig categories, in the same way as $\mathbb{N}$ is initial in the category of rigs. As a by-product, an explicit description of the homomorphisms of rig categories from a suitable version of $\widehat{\mathbb{F}\mathbb{S} et}$ into any (semistrict) rig category $\mathbb{S}$ is obtained in terms of a sequence of automorphisms of the objects $1+\stackrel{n)}{\cdots}+1$ in $\mathbb{S}$ for each $n\geq 0$.

math.CT↗
arXiv2016

On the representations of 2-groups in {Baez-Crans} 2-vector spaces

We prove that the theory of representations of a finite 2-group $\mathbb{G}$ in Baez-Crans 2-vector spaces over a field $k$ of characteristic zero essentially reduces to the theory of $k$-linear representations of the group of isomorphism classes of objects of $\mathbb{G}$, the remaining homotopy invariants of $\mathbb{G}$ playing no role. It is also argued that a similar result is expected to hold for topological representations of compact topological 2-groups in suitable topological Baez-Crans 2-vector spaces.

math.CT↗
arXiv2013

Permutation 2-groups I: structure and splitness

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a one-object 2-group. By a permutation 2-group we mean the 2-group $\mathbb{S}ym(\mathcal{G})$ of self-equivalences of a groupoid $\mathcal{G}$ and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the symmetric groups $\mathsf{S}_n$, $n\geq 1$, obtained when $\mathcal{G}$ is a finite discrete groupoid. After introducing the wreath 2-product $\mathsf{S}_n\wr\wr\ \mathbb{G}$ of the symmetric group $\mathsf{S}_n$ with an arbitrary 2-group $\mathbb{G}$, it is shown that for any (finite type) groupoid $\mathcal{G}$ the permutation 2-group $\mathbb{S}ym(\mathcal{G})$ is equivalent to a product of wreath 2-products of the form $\mathsf{S}_n\wr\wr\ \mathbb{S}ym(\mathcal{B}\mathsf{G})$, where $\mathcal{B}\mathsf{G}$ is the delooping of $\mathsf{G}$. This is next used to compute the homotopy invariants of $\mathbb{S}ym(\mathcal{G})$ which classify it up to equivalence. In particular, we prove that $\mathbb{S}ym(\mathcal{G})$ can be non-split, and that the step from the trivial groupoid $\mathcal{B}\mathsf{1}$ to an arbitrary one-object groupoid $\mathcal{B}\mathsf{G}$ is in fact the only source of non-splitness. Various examples of permutation 2-groups are explicitly computed, in particular the permutation 2-group of the underlying groupoid of a (finite type) 2-group. It also follows from well known results about the symmetric groups that the permutation 2-group of the groupoid of all finite sets and bijections between them is equivalent to the direct product 2-group $\mathbb{Z}_2[1]\times\mathbb{Z}_2[0]$, where $\mathbb{Z}_2[0]$ and $\mathbb{Z}_2[1]$ stand for the group $\mathbb{Z}_2$ thought of as a discrete and a one-object 2-group, respectively.

math.CT↗
arXiv2010

The 2-group of symmetries of a split chain complex

We explicitly compute the 2-group of self-equivalences and (homotopy classes of) chain homotopies between them for any {\it split} chain complex $A_{\bullet}$ in an arbitrary $\kb$-linear abelian category ($\kb$ any commutative ring with unit). In particular, it is shown that it is a {\it split} 2-group whose equivalence class depends only on the homology of $A_{\bullet}$, and that it is equivalent to the trivial 2-group when $A_\bullet$ is a split exact sequence. This provides a description of the {\it general linear 2-group} of a Baez and Crans 2-vector space over an arbitrary field $\mathbb{F}$ and of its generalization to chain complexes of vector spaces of arbitrary length.

math.KT↗
arXiv2009

On the regular representation of an (essentially) finite 2-group

The regular representation of an essentially finite 2-group $\mathbb{G}$ in the 2-category $\mathbf{2Vect}_k$ of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in $\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})$ are 2-vector spaces under quite standard assumptions on the field $k$, and a formula giving the corresponding "intertwining numbers" is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor ${\boldmath$ω$}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k$ is representable with the regular representation as representing object. As a consequence we obtain a $k$-linear equivalence between the 2-vector space $\mathbf{Vect}_k^{\mathcal{G}}$ of functors from the underlying groupoid of $\mathbb{G}$ to $\mathbf{Vect}_k$, on the one hand, and the $k$-linear category $\mathcal{E} nd({\boldmath$ω$})$ of pseudonatural endomorphisms of ${\boldmath$ω$}$, on the other hand. We conclude that $\mathcal{E} nd({\boldmath$ω$})$ is a 2-vector space, and we (partially) describe a basis of it.

math.CT↗
arXiv2006

Generalized 2-vector spaces and general linear 2-groups

In this paper a notion of {\it generalized 2-vector space} is introduced which includes Kapranov and Voevodsky 2-vector spaces. Various kinds of generalized 2-vector spaces are considered and examples are given. The existence of non free generalized 2-vector spaces and of generalized 2-vector spaces which are non Karoubian (hence, non abelian) categories is discussed, and it is shown how any generalized 2-vector space can be identified with a full subcategory of an (abelian) functor category with values in the category ${\bf VECT}_K$ of (possibly infinite dimensional) vector spaces. The corresponding general linear 2-groups $\mathbb{G}\mathbb{L}({\bf Vect}_K[\mathcal{C}])$ are considered. Specifically, it is shown that $\mathbb{G}\mathbb{L}({\bf Vect}_K[\mathcal{C}])$ always contains as a (non full) sub-2-group the 2-group ${\sf Equiv}_{Cat}(\mathcal{C})$ (hence, for finite categories $\mathcal{C}$, they contain {\sl Weyl sub-2-groups} analogous to usual Weyl subgroups of the general linear groups), and $\mathbb{G}\mathbb{L}({\bf Vect}_K[\mathcal{C}])$ is explicitly computed (up to equivalence) in a special case of generalized 2-vector spaces which include those of Kapranov and Voevodsky. Finally, other important drawbacks of the notion of generalized 2-vector space, besides the fact that it is in general a non Karoubian category, are also mentioned at the end of the paper.

math.CT↗
arXiv2005

Representation theory of 2-groups on finite dimensional 2-vector spaces

In this paper, the 2-category $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$ of (weak) representations of an arbitrary (weak) 2-group $\mathbb{G}$ on (some version of) Kapranov and Voevodsky's 2-category of (complex) 2-vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invariants $π_0(\mathbb{G})$, $π_1(\mathbb{G})$ and $[α]\in H^3(π_0(\mathbb{G}),π_1(\mathbb{G}))$ classifying $\mathbb{G}$. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain the the {\it monoidal} category of linear representations (more generally, the category of $[z]$-projective representations, for any given cohomology class $[z]\in H^2(π_0(\mathbb{G}),\mathbb{C}^*)) of the first homotopy group $π_0(\mathbb{G})$ as well as its category of representations on finite sets both live in $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$, the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of morphisms between suitable 1-dimensional representations) and the second as a subcategory of the homotopy category of $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$.

math.CT↗
arXiv2004

2-cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudofunctors

In this paper we take up again the deformation theory for $K$-linear pseudofunctors initiated in a previous work (Adv. Math. 182 (2004) 204-277). We start by introducing a notion of a 2-cosemisimplicial object in an arbitrary 2-category and analyzing the corresponding coherence question, where the permutohedra make their appearence. We then describe a general method to obtain cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in the 2-category ${\bf Cat}_K$ of small $K$-linear categories and prove that the deformation complex introduced in the above mentioned work can be obtained by this method from a 2-cosemisimplicial object that can be associated to the pseudofunctor. Finally, using a generalization to the context of $K$-linear categories of the deviation calculus introduced by Markl and Stasheff for $K$-modules (J. Algebra 170 (1994) 122), it is shown that the obstructions to the integrability of an $n^{th}$-order deformation of a pseudofunctor indeed correspond to cocycles in the third cohomology group, a question which remained open in our previous work.

math.QA↗
arXiv2002

Cohomology and deformation theory of monoidal 2-categories I

We define a cohomology for an arbitrary $K$-linear semistrict semigroupal 2-category $(\mathfrak{C},\otimes)$ (called in the paper a Gray semigroup) and show that its first order (unitary) deformations, up to the suitable notion of equivalence, are in one-one correspondence with the elements of the second cohomology group. Fundamental to the construction is a double complex, similar to Gerstenhaber-Schack's double complex for bialgebras. We also identify the cohomologies describing separately the deformations of the tensor product, the associator and the pentagonator. To obtain these results, a cohomology theory for an arbitrary $K$-linear unitary pseudofunctor is introduced describing its purely pseudofunctorial deformations, and generalizing Yetter's cohomology for semigroupal functors. The corresponding higher order obstructions will be considered in a future paper.

math.QA↗