Theory and applications of categories

Vol. 31, num. 32, p. 907-927

Date of publication: 2016-10-06

Abstract:

We study the theory of representations of a 2-group G in Baez-Crans 2-vector spaces over a field k of arbitrary characteristic, and the corresponding 2-vector spaces of intertwiners. We also characterize the irreducible and indecomposable representations. Finally, it is shown that when the 2-group is finite and the base field k is of characteristic zero or coprime to the orders of the homotopy groups of G, the theory essentially reduces to the theory of k-linear representations of the first homotopy group of G, the remaining homotopy invariants of G playing no role.]]>

Abstract:

This project focuses on various aspects of the areas of algebraic geometry, symplectic geometry, commutative algebra, algebraic topology and their interactions, and their applications to biology, physics and robotics. We propose to approach the problems considered both from the traditional view of research in mathematics and from a multidisciplinary perspective on the part of applications. This project is the continuation of MTM2012-38122-C03-01, which has been very successful and has had a major impact both within the mathematical community and other disciplines and even outside the scientific community (to name a few indicators, we have 160 publications since 2005, the group's work has received more than 700 citations according to Scopus and there are members of the group with h-index between 6 and 8). The current group has 23 researchers; 15 of them doctors (of which 11 are senior and 4 are postdoc) and 8 tPhD students. In the last decades, algebraic geometry has experimented a spectacular development in interaction with affine areas of research. For instance, some ideas of algebraic geometry find its generalization in symplectic geometry; in Poisson Geometry, the algebraic techniques are omnipresent in the study of generalized complex geometry also the algebraic topology techniques are present in the modern homotopy theory of schemes. Other examples of interaction of disciplines is observed in the recent applications of geometric analysis techniques to the realm of Poisson Geometry and Complex geometry. In this project we will keep on exploring the connection among these areas and we will endeavour to solve important problems relative to known conjectures such as Xiaos conjectures for irregular varieties in Algebraic Geometry, Voevodskys nilpotence conjecture in K-theory or the conjecture of Guillemin-Sternberg concerning quantization and reduction for symplectic manifolds and their generalizations, or the Sturmfels-Sullivant conjecture on phylogenetic varieties. Given the success of the results obtained in the preceding project, in this project we plan to consolidate the interdisciplinary aspects of the project. A strong mathematical component is observed in the publications of the group in the areas of biomathematics, robotics and physics. Notwithstanding, those results can really be applied to those disciplines as observed in the list of high-impact journals where these results are published (Nature Methods, Systematic Biology, Molecular Biology and Evolution, BMC Evolutionary Biology, International Journal of Computer vision, Physical Review Letters, Journal of Cosmology and Astroparticle Physics, Physical Review D, Physical Reviews E). Our team collaborates with various national and international groups. We enclose details of the collaborators in the scientific proposal and highlighting now the following national collaborative centers: ICMAT, IRI, Centre for Genomic Regulation, UAB and UB and international: MIT-Northeastern, U. Pavia, U. Bayreuth, U.Kansas, U .Leicester, KU Leuven, UC Berkeley, Mathematical Institute of the Polish Academy of Sciences, U. Tasmania, New Zealand Biomathematics Research Centre.]]>

Date: 2015-02-01

Abstract:

Col.lecció de problemes apareguts en diferents actes d’avaluació de l’assignatura Fonaments Matemàtics del Grau en Enginyeria Informàtica de la Facultat d’Informàtica de Barcelona, U.P.C., des del setembre de 2010.]]>

Advances in mathematics

Vol. 258, p. 286-350

DOI: 10.1016/j.aim.2014.03.011

Date of publication: 2014-06-20

Abstract:

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 Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the symmetric groups S-n, n >= 1, obtained when G is a finite discrete groupoid.; After introducing the wreath 2-product S-n (sic) G of the symmetric group S-n with an arbitrary 2-group G, it is shown that for any (finite type) groupoid G the permutation 2-group Sym(G) is equivalent to a product of wreath 2-products of the form S-n (sic) Sym(G) for a group G thought of as a one-object groupoid. This is next used to compute the homotopy invariants of Sym(G) which classify it up to equivalence. Using a previously shown splitness criterion for strict 2-groups, it is then proved that Sym(G) can be non-split, and that the step from the trivial groupoid to an arbitrary one-object groupoid is 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 Z(2)[1] x Z(2)[0] where Z(2)[0] and Z(2)[1] stand for the group Z(2) thought of as a discrete and a one-object 2-group, respectively. (C) 2014 Elsevier Inc. All rights reserved.]]>

Journal of algebra

Vol. 351, num. 1, p. 319-349

DOI: 10.1016/j.jalgebra.2011.11.016

Date of publication: 2012-02

Advances in mathematics

Vol. 227, num. 1, p. 170-209

DOI: 10.1016/j.aim.2011.01.016

Date of publication: 2011-05-01

Abstract:

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 ${\boldmath$\omega$}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k$ 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 $\mathcal{E} nd({\boldmath$\omega$})$ of pseudonatural endomorphisms of ${\boldmath$\omega$}$, on the other hand. We conclude that $\mathcal{E} nd({\boldmath$\omega$})$ is a 2-vector space, and we (partially) describe a basis of it.]]>

Date: 2010-12-15

Abstract:

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.]]>

Date: 2010-12-01

Abstract:

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 ${\boldmath$\omega$}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k$ 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 $\mathcal{E} nd({\boldmath$\omega$})$ of pseudonatural endomorphisms of ${\boldmath$\omega$}$, on the other hand. We conclude that $\mathcal{E} nd({\boldmath$\omega$})$ is a 2-vector space, and we (partially) describe a basis of it.]]>

Journal of pure and applied algebra

Vol. 191, num. -, p. 223-264

Date of publication: 2008-06

Advanced Course on Simplicial Methods in Higher Categories

Presentation's date: 2008-03-03

Homotopy structures in geometry and algebra; derived categories, higher categories

Presentation's date: 2008-02-02

V Seminar on Categories and Applications

Presentation's date: 2008-01-01

Advances in mathematics

Vol. 213, num. 1, p. 53-92

Date of publication: 2007-08

Mathematical proceedings of the Cambridge Philosophical Society

Vol. 142, num. 3, p. 407-428

Date of publication: 2007-05

IV Seminar on Categories and Applications

Presentation's date: 2007-01-01

Date: 2007-01

International Category Theory Conference

Presentation's date: 2006-06-30

Date: 2004-09

Journal of pure and applied algebra

Vol. 191, num. 3, p. 223-264

Date of publication: 2004-08

Advances in mathematics

Vol. 182, p. 204-277

Date of publication: 2004-03

Summer Program on n-categories, Foundations and Applications

Presentation's date: 2004-01-01

Workshop on Categorification and Higher-order Geometry

Presentation's date: 2003-01-01

Date: 2001-04