Mapping spaces in Quasi-categories

We apply the Dwyer-Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [DS1], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way

Rigidification of quasi-categories

We give a new construction for rigidifying a quasi-category into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets obtained by stringing simplices together. As an application of these methods, we use our model to reprove some basic facts from Lurie's "Higher Topos Theory" regarding the rigidification process.Comment: 26 page

Database queries and constraints via lifting problems

Previous work has demonstrated that categories are useful and expressive models for databases. In the present paper we build on that model, showing that certain queries and constraints correspond to lifting problems, as found in modern approaches to algebraic topology. In our formulation, each so-called SPARQL graph pattern query corresponds to a category-theoretic lifting problem, whereby the set of solutions to the query is precisely the set of lifts. We interpret constraints within the same formalism and then investigate some basic properties of queries and constraints. In particular, to any database $\pi$ we can associate a certain derived database \Qry(\pi) of queries on $\pi$. As an application, we explain how giving users access to certain parts of \Qry(\pi), rather than direct access to $\pi$, improves ones ability to manage the impact of schema evolution

What makes men leak? An investigation of objective and self-report measures of urinary incontinence early after radical prostatectomy

AimsPelvic floor muscle training for patients having radical prostatectomy promotes contraction of these muscles in anticipation of activities that may provoke urine leakage. The aims of this study were: to determine the contribution of the individual activities comprising a standardised 1-hour pad test (1HPT) to overall urine leakage early after radical prostatectomy; and to investigate relationships between the 1HPT, 24-hour pad test (24HPT) and the International Consultation on Incontinence QuestionnaireShort Form (ICIQ-SF) early after radical prostatectomy. MethodsA prospective analysis of patients having radical prostatectomy and receiving pelvic floor muscle training (n=33). Participants completed the 1HPT, 24HPT and ICIQ-SF at 3 and 6 weeks postoperatively. Participants wore a separate, pre-weighed continence pad for each of the seven activities comprising the 1HPT; pads were weighed separately and together to calculate activity-related and overall urine leakage. ResultsWalking at a comfortable speed and drinking while sitting were the two activities contributing most to overall urine leakage, albeit these activities also comprised 75% of 1HPT time. All component activities contributed a minimum 75% of overall urine leakage. There were significant and strong to very strong correlations between all of the 1HPT, 24HPT, and ICIQ-SF at 3 weeks postoperatively. There were significant decreases in 24HPT (P=0.032) and ICIQ-SF (P=0.001) but no significant change in 1HPT from 3 to 6 weeks postoperatively. ConclusionsPelvic floor muscle training should include contraction of these muscles in sedentary and walking postures. The 1HPT correlates well with the 24HPT, but may not be sensitive to early postoperative improvements in urinary leakage. Neurourol. Urodynam. 35:225-229, 2016. (c) 2014 Wiley Periodicals, Inc

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.Comment: Final version, to appear in Israel J. Mat

Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories

(Co)Simplicial Descent Categories

In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natural structure of Brown category of cofibrant objects on sD. We use these facts to produce cofiber sequences on the localized category of D by E, which give rise to a natural Verdier triangulated structure in the stable case.Comment: Final version. To appear in the J. Pure Appl. Algebr

Two-dimensional enrichment analysis for mining high-level imaging genetic associations

Enrichment analysis has been widely applied in the genome-wide association studies (GWAS), where gene sets corresponding to biological pathways are examined for significant associations with a phenotype to help increase statistical power and improve biological interpretation. In this work, we expand the scope of enrichment analysis into brain imaging genetics, an emerging field that studies how genetic variation influences brain structure and function measured by neuroimaging quantitative traits (QT). Given the high dimensionality of both imaging and genetic data, we propose to study Imaging Genetic Enrichment Analysis (IGEA), a new enrichment analysis paradigm that jointly considers meaningful gene sets (GS) and brain circuits (BC) and examines whether any given GS-BC pair is enriched in a list of gene-QT findings. Using gene expression data from Allen Human Brain Atlas and imaging genetics data from Alzheimer's Disease Neuroimaging Initiative as test beds, we present an IGEA framework and conduct a proof-of-concept study. This empirical study identifies 12 significant high level two dimensional imaging genetics modules. Many of these modules are relevant to a variety of neurobiological pathways or neurodegenerative diseases, showing the promise of the proposal framework for providing insight into the mechanism of complex diseases

Measures of Autozygosity in Decline: Globalization, Urbanization, and Its Implications for Medical Genetics

This research investigates the influence of demographic factors on human genetic sub-structure. In our discovery cohort, we show significant demographic trends for decreasing autozygosity associated with population variation in chronological age. Autozygosity, the genomic signature of consanguinity, is identifiable on a genome-wide level as extended tracts of homozygosity. We identified an average of 28.6 tracts of extended homozygosity greater than 1 Mb in length in a representative population of 809 unrelated North Americans of European descent ranging in chronological age from 19–99 years old. These homozygous tracts made up a population average of 42 Mb of the genome corresponding to 1.6% of the entire genome, with each homozygous tract an average of 1.5 Mb in length. Runs of homozygosity are steadily decreasing in size and frequency as time progresses (linear regression, p<0.05). We also calculated inbreeding coefficients and showed a significant trend for population-wide increasing heterozygosity outside of linkage disequilibrium. We successfully replicated these associations in a demographically similar cohort comprised of a subgroup of 477 Baltimore Longitudinal Study of Aging participants. We also constructed statistical models showing predicted declining rates of autozygosity spanning the 20th century. These predictive models suggest a 14.0% decrease in the frequency of these runs of homozygosity and a 24.3% decrease in the percent of the genome in runs of homozygosity, as well as a 30.5% decrease in excess homozygosity based on the linkage pruned inbreeding coefficients. The trend for decreasing autozygosity due to panmixia and larger effective population sizes will likely affect the frequency of rare recessive genetic diseases in the future. Autozygosity has declined, and it seems it will continue doing so