Enriched weakness

The basic notions of category theory, such as limit, adjunction, and orthogonality, all involve assertions of the existence and uniqueness of certain arrows. Weak notions arise when one drops the uniqueness requirement and asks only for existence. The enriched versions of the usual notions involve certain morphisms between hom-objects being invertible; here we introduce enriched versions of the weak notions by asking that the morphisms between hom-objects belong to a chosen class of "surjections". We study in particular injectivity (weak orthogonality) in the enriched context, and illustrate how it can be used to describe homotopy coherent structures.Comment: 25 pages; v2 minor changes, to appear in JPA

An embedding theorem for adhesive categories

Adhesive categories are categories which have pushouts with one leg a monomorphism, all pullbacks, and certain exactness conditions relating these pushouts and pullbacks. We give a new proof of the fact that every topos is adhesive. We also prove a converse: every small adhesive category has a fully faithful functor in a topos, with the functor preserving the all the structure. Combining these two results, we see that the exactness conditions in the definition of adhesive category are exactly the relationship between pushouts along monomorphisms and pullbacks which hold in any topos.Comment: 8 page

Enhanced 2-categories and limits for lax morphisms

We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is done using the framework of 2-monads. In order to characterize the limits which exist in this context, we need to consider also the functors which do strictly preserve the extra structure. We show how such a 2-category of weak morphisms which is "enhanced", by specifying which of these weak morphisms are actually strict, can be thought of as category enriched over a particular base cartesian closed category F. We give a complete characterization, in terms of F-enriched category theory, of the limits which exist in such 2-categories of categories with extra structure.Comment: 77 pages; v2 minor changes only, to appear in Advance

Morita contexts as lax functors

Monads are well known to be equivalent to lax functors out of the terminal category. Morita contexts are here shown to be lax functors out of the chaotic category with two objects. This allows various aspects in the theory of Morita contexts to be seen as special cases of general results about lax functors. The account we give of this could serve as an introduction to lax functors for those familiar with the theory of monads. We also prove some very general results along these lines relative to a given 2-comonad, with the classical case of ordinary monad theory amounting to the case of the identity comonad on Cat.Comment: v2 minor changes, added references; to appear in Applied Categorical Structure

Good work, little soldier: Text and pretext

This article reads the relation between Denis's Beau Travail and Jean-Luc Godard's 1960 film Le Petit Soldat as a film-on-film variant of film-on-book adaptation. The model informing this reading is not so much intertextual as pretextual. The principal points of contact between the two films discussed are 'actor' (Michel Subor), 'character' (Bruno Forestier) and 'narrator' (Forestier/Galoup). The use in Beau Travail of Le Petit Soldat is compared with and differentiated from the use of Melville's 'Billy Budd, Sailor'. The conclusion arrived at is that the film-on-film relation can be read as a development of the mirror motif borrowed from Godard by Denis, in order to replace abyssal models of intertextual infinity with the finitudes of abyssal reflexivity. This is to offer a model of pretextuality that is not dependent on privileging the pretext: implicit is the suggestion that Beau Travail and Le Petit Soldat may be read as a single, if hybrid, text

The point in time: Precise chronology in early godard

This essay considers the significance of reference to chronologically specific material (including newspapers, magazines, radio broadcasts, graffiti) in Godard’s first three films of the 1960s - Le Petit soldat/The Little Soldier (made 1960, released 1963), Une femme est une femme/A Woman is a Woman (1961) and Vivre sa vie/My Life to Live (1962), and in Pierrot le fou/Crazy Pete (1965) - arguing that such ephemeral traces of a period can serve as a means of access to the political import of these films, and also are part of a larger concern in Godard with questions of time and history, questions that he is still asking thirty or forty years later, in works like Histoire(s) du cinéma/Histories of Cinema (1998) and Éloge de l’amour/In Praise of Love (2001). © Intellect Ltd 2003

Sprawling by the Lake: How IDA-Granted Property Tax Exemptions Undermine Older Parts of the Buffalo/Niagara Metro Area

An examination of the geographic distribution of property tax exemptions given to businesses in 2005 by the nine state-regulated Industrial Development Agencies (IDAs) in the Buffalo/Niagara metro area reveals they have subsidized job creation outside of the region’s oldest, most densely populated and most transit-accessible areas, despite the fact those areas are most in need of jobs and reinvestment. The exemptions’ sprawling, pro-suburban bias is especially evident in Erie County and far less problematic in Niagara

Icons

Categorical orthodoxy has it that collections of ordinary mathematical structures such as groups, rings, or spaces, form categories (such as the category of groups); collections of 1-dimensional categorical structures, such as categories, monoidal categories, or categories with finite limits, form 2-categories; and collections of 2-dimensional categorical structures, such as 2-categories or bicategories, form 3-categories. We describe a useful way in which to regard bicategories as objects of a 2-category. This is a bit surprising both for technical and for conceptual reasons. The 2-cells of this 2-category are the crucial new ingredient; they are the icons of the title. These can be thought of as ``the oplax natural transformations whose components are identities'', but we shall also give a more elementary description. We describe some properties of these icons, and give applications to monoidal categories, to 2-nerves of bicategories, to 2-dimensional Lawvere theories, and to bundles of bicategories.Comment: 23 page