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.

Finiteness obstructions and Euler characteristics of categories

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Robust discrimination between EEG responses to categories of environmental sounds in early coma.

Humans can recognize categories of environmental sounds, including vocalizations produced by humans and animals and the sounds of man-made objects. Most neuroimaging investigations of environmental sound discrimination have studied subjects while consciously perceiving and often explicitly recognizing the stimuli. Consequently, it remains unclear to what extent auditory object processing occurs independently of task demands and consciousness. Studies in animal models have shown that environmental sound discrimination at a neural level persists even in anesthetized preparations, whereas data from anesthetized humans has thus far provided null results. Here, we studied comatose patients as a model of environmental sound discrimination capacities during unconsciousness. We included 19 comatose patients treated with therapeutic hypothermia (TH) during the first 2 days of coma, while recording nineteen-channel electroencephalography (EEG). At the level of each individual patient, we applied a decoding algorithm to quantify the differential EEG responses to human vs. animal vocalizations as well as to sounds of living vocalizations vs. man-made objects. Discrimination between vocalization types was accurate in 11 patients and discrimination between sounds from living and man-made sources in 10 patients. At the group level, the results were significant only for the comparison between vocalization types. These results lay the groundwork for disentangling truly preferential activations in response to auditory categories, and the contribution of awareness to auditory category discrimination.

Dukes B colorectal cancer: distinct genetic categories and clinical outcome based on proximal or distal tumor location.

PURPOSE: The aim of this study was to determine whether tumor location proximal or distal to the splenic flexure is associated with distinct molecular patterns and can predict clinical outcome in a homogeneous group of patients with Dukes B (T3-T4, N0, M0) colorectal cancer. It has been hypothesized that proximal and distal colorectal cancer may arise through different pathogenetic mechanisms. Although p53 and Ki-ras gene mutations occur frequently in distal tumors, another form of genomic instability associated with defective DNA mismatch repair has been predominantly identified in the proximal colon. To date, however, the clinical usefulness of these molecular characteristics remains unproven. METHODS: A total of 126 patients with a lymph node-negative sporadic colon or rectum adenocarcinoma were prospectively assessed with the endpoint of death by cancer. No patient received either radiotherapy or chemotherapy. p53 protein was studied by immunohistochemistry using DO-7 monoclonal antibody, and p53 and Ki-ras gene mutations were detected by single strand conformation polymorphism assay. RESULTS: During a mean follow-up of 67 months, the overall five-year survival was 70 percent. Nuclear p53 staining was found in 57 tumors (47 percent), and was more frequent in distal than in proximal tumors (55 vs. 21 percent; chi-squared test, P < 0.001). For the whole group, p53 protein expression correlated with poor survival in univariate and multivariate analysis (log-rank test, P = 0.01; hazard ratio = 2.16; 95 percent confidence interval = 1.12-4.11, P = 0.02). Distal colon tumors and rectal tumors exhibited similar molecular patterns and showed no difference in clinical outcome. In comparison with distal colorectal cancer, proximal tumors were found to be statistically significantly different on the following factors: mucinous content (P = 0.008), degree of histologic differentiation (P = 0.012), p53 protein expression, and gene mutation (P = 0.001 and 0.01 respectively). Finally, patients with proximal tumors had a marginally better survival than those with distal colon or rectal cancers (log-rank test, P = 0.045). CONCLUSION: In this series of Dukes B colorectal cancers, p53 protein expression was an independent factor for survival, which also correlated with tumor location. Eighty-six percent of p53-positive tumors were located in the distal colon and rectum. Distal colon and rectum tumors had similar molecular and clinical characteristics. In contrast, proximal neoplasms seem to represent a distinct entity, with specific histopathologic characteristics, molecular patterns, and clinical outcome. Location of the neoplasm in reference to the splenic flexure should be considered before group stratification in future trials of adjuvant chemotherapy in patients with Dukes B tumors.

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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Diarrheagenic Escherichia coli categories among the traditional enteropathogenic E. coli O serogroups: a review

The socalled enteropathogenic Escherichia coli (EPEC) O serogroups include typical and atypical EPEC, enterohaemorrragic E. coli, enterotoxigenic E. coli, and enteroaggregative E. coli. The aim of this article is to review the composition of each O serogroup and the major serotypes, clones, and additional virulence characteristics of each of these diarrheageniccategories. Their adherence patterns and genetic relationships are also presented. The review is based on the study of 805 strains of serogroups O26, O55, O86, O111, O114, O119, O125, O126, O1127, O128, and O142 most of which isolated in São Paulo from children with diarrhea between 1970 and 1990. Since some O serogroups include more than one diarrheageniccategory O serogrouping only should be abandoned as a diagnostic method. However serotyping is a reliable method for those serotypes that correspond to clones.