A curious example of two model categories and some associated differential graded algebras

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Homotopy limits for 2-categories

We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

Model structures on the category of small double categories

In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofi brant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves and categorifications.

Derived categories and Grothendieck duality

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Cluster algebras, quiver representations and triangulated categories

This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria)and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.

On the homotopy theory of spectral categories

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A Thomason model structure on the category of small n-fold categories

We construct a cofibrantly generated Thomason model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences.

The Argentine ant (Linepithema humile) invasion effects on Mediterranean forests’ trophic web: blue tit (Parus caeruleus) reproduction and development

Les invasions biològiques representen una greu amenaça per al funcionament dels ecosistemes i per a la preservació de la biodiversitat.. La formiga argentina (Linepithema humile) està considerada com una de les 100 espècies invasores més nocives. Prospera en extenses àrees de clima mediterrani de regions temperades i subtropicals de tots els continents amb l’excepció de l’Antàrtida. És una formiga dominant i una competidora agressiva que mitjançant múltiples mecanismes, des de predació directe a competència, produeix efectes negatius en una amplia varietat de taxons, principalment formigues i altres artròpodes, però també vertebrats. S’ha investigat, per primera vegada, els efectes de la formiga invasiva sobre les comunitats d’artròpodes de fullatge i com aquestes pertorbacions es transmeten en la xarxa tròfica del bosc esclerofil•le mediterrani. En les suredes estudiades la invasió de formiga argentina és causa directe de la extinció local de la gran majoria de poblacions de formigues natives. En el període mostrejat s’han constatat també impactes negatius en la diversitat i en l’abundància d’artròpodes natius en les capçades dels arbres, particularment d’erugues. Una avaluació preliminar basada únicament amb dades del 2005 indica que, reduint la disponibilitat d’erugues, la formiga argentina empobreix l’hàbitat reproductiu de la mallerenga blava (Parus caeruleus). La mallerenga blava basa la dieta insectívora estricte de la seva pollada fonamentalment en les erugues. No hem detectat impactes en l’èxit reproductiu de les mallerengues blaves en zones envaïdes. Els polls crescuts en àrees envaïdes assoleixen una condició física similar als de les zones no envaïdes, però la reducció en la disponibilitat d’erugues associada a la invasió de formiga argentina es tradueix en un creixement descompassat i en una menor mida estructural del polls volanders. Així, les pertorbacions en la comunitat d’artròpodes associades a la invasió de la formiga argentina promouen efectes bottom-up que acaben perjudicant el desenvolupament dels polls de mallerenga blava.

Finiteness obstructions and Euler characteristics of categories

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Monads in double categories

We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and dene what it means for a double category to admit the construction of free monads. Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads.

Euler characteristics of categories and homotopy colimits

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Spectrum of some trianguled categories

In this paper, we compute the triangular spectrum (as de fined by P. Balmer) of two classes of tensor triangulated categories which are quite common in algebraic geometry. One of them is the derived category of G-equivariant sheaves on a smooth scheme X, for a fi nite group G. The other class is the derived category of split superschemes.

Multiple correspondence analysis of a subset of response categories

In the analysis of multivariate categorical data, typically the analysis of questionnaire data, it is often advantageous, for substantive and technical reasons, to analyse a subset of response categories. In multiple correspondence analysis, where each category is coded as a column of an indicator matrix or row and column of Burt matrix, it is not correct to simply analyse the corresponding submatrix of data, since the whole geometric structure is different for the submatrix . A simple modification of the correspondence analysis algorithm allows the overall geometric structure of the complete data set to be retained while calculating the solution for the selected subset of points. This strategy is useful for analysing patterns of response amongst any subset of categories and relating these patterns to demographic factors, especially for studying patterns of particular responses such as missing and neutral responses. The methodology is illustrated using data from the International Social Survey Program on Family and Changing Gender Roles in 1994.

Subset correspondence analysis: Visualizing relationships among a selected set of response categories from a questionnaire survey

It is shown how correspondence analysis may be applied to a subset of response categories from a questionnaire survey, for example the subset of undecided responses or the subset of responses for a particular category. The idea is to maintain the original relative frequencies of the categories and not re-express them relative to totals within the subset, as would normally be done in a regular correspondence analysis of the subset. Furthermore, the masses and chi-square metric assigned to the data subset are the same as those in the correspondence analysis of the whole data set. This variant of the method, called Subset Correspondence Analysis, is illustrated on data from the ISSP survey on Family and Changing Gender Roles.