Obstruction Theory in Model Categories

Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.Comment: 17 pages. v3 includes improved introduction and several other minor improvement

Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories

The Leander McCormick Observatory of the University of Virginia

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The Ice Palace at St. Paul

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The Sturgeon Fishery

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A comparison of moored and free-drifting sediment traps of two different designs

The Kiel conical sediment trap and a 3:1 right cylinder were simultaneously deployed in both free-drifting and moored modes on four separate occasions over the Peruvian shelf in order to compare downward flux measurements derived from each…

Duality and Pro-Spectra

Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier-Whitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-34.abs.htm

Invited Paper. The Hermeneutic Wager: Building Community in Pediatric Neuro-Oncology

During the Covid-19 pandemic, Hovey was contacted by the lead of a pan-Canadian working group on pediatric brain tumours (PBTWG). While all stakeholders (researchers, clinicians, regulators, patient advocates, ethicists, and industry experts) were highly motivated to address barriers through innovative strategies in collaboration, clinical research, regulation, and business models, advancement has been challenging on multiple levels. Hovey and his team were tasked to facilitate and successfully engage this diverse divisive group of stakeholders to achieve their goals. Inspired by Richard Kearney’s anatheistic wager, the hermeneutic wager acts simultaneously as a team building and research approach, as it serves to gain insight into the perspectives of members of a purposeful community. Through its five conversations, namely imagination, humility, commitment, discernment, and hospitality, the hermeneutic wager elicits responses from the participants that are based on meaningful participation in a relational approach of community co-creation. We individually interviewed the PBTWG facilitators (5). With informed consent, our research team also recorded all 5 of the PBTWG work group meetings (20 participants from 6 stakeholder groups) and break-out room meetings and took notes which consist of rich and extensive narrative data. This data was analyzed alongside the individual PBTWG interviews. The audio and visual data collected via a secure Zoom platform was then transcribed verbatim and analyzed interpretively according to the applied philosophical hermeneutic tradition. Findings centered around six points: “The Work of Stories,” “Changing Landscapes: Community / Communication not Consensus,” “Let the Words Lead You,” “Those Words Matter,” “Metaphors as a Bridge to Understanding,” and “A Road Map to be Inspired By.” Through these findings, we contend that the hermeneutic wager is an invitation for conversation that builds a path to the generation of new and creative understandings that transcend previous ways of knowing. The efficacy of the hermeneutic wager resides in its ability to help build a community of people who work together through and across difference to arrive at a shared understanding and collective outcome. &nbsp

Homotopy Theoretic Models of Type Theory

We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it. On the other hand, those conditions are easy to check and provide a wide class of models some of which are listed in the paper.Comment: Corrected version of the published articl