Algebraic weak factorisation systems II: categories of weak maps

We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of "homotopy category", that freely adjoins a section for every "acyclic fibration" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of spans with left leg an acyclic fibration. We moreover show that the 2-functor sending each AWFS on a suitable category to its cofibrant replacement comonad has a fully faithful right adjoint: so exhibiting the theory of comonads, and dually of monads, as incorporated into the theory of AWFS. We also describe various applications of the general theory: to the generalised sketches of Kinoshita-Power-Takeyama, to the two-dimensional monad theory of Blackwell-Kelly-Power, and to the theory of dg-categories.Comment: 30 pages, final journal versio

Two-dimensional regularity and exactness

We define notions of regularity and (Barr-)exactness for 2-categories. In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects, injective on objects and fully faithful), as (bijective on objects, fully faithful), and as (bijective on objects and full, faithful). The correctness of our notions is justified using the theory of lex colimits introduced by Lack and the second author. Along the way, we develop an abstract theory of regularity and exactness relative to a kernel--quotient factorisation, extending earlier work of Street and others.Comment: 37 page

What is conservatism? History, ideology and party

Is there a political philosophy of conservatism? A history of the phenomenon written along sceptical lines casts doubt on the existence of a transhistorical doctrine, or even an enduring conservative outlook. The main typologies of conservatism uniformly trace its origins to opposition to the French Revolution. Accordingly, Edmund Burke is standardly singled out as the ‘father’ of this style of politics. Yet Burke was de facto an opposition Whig who devoted his career to assorted programmes of reform. In restoring Burke to his original milieu, the argument presented here takes issue with 20th-century accounts of conservative ideology developed by such figures as Karl Mannheim, Klaus Epstein and Samuel Huntington. It argues that the idea of a conservative tradition is best seen as a belated construction, and that the notion of a univocal philosophy of conservatism is basically misconceived. </jats:p

REFLECTIONS ON THE POLITICAL THOUGHT OF THE IRISH REVOLUTION

ABSTRACTExamining the political thought of the Irish Revolution poses two distinct problems. First, we need to establish how we should date the Revolution for the purposes of intellectual history. There is no doubting that the 1916 Easter Rising was an event in British and Irish politics, but it was also an event in the world of ideas. Any serious consideration of this episode and its aftermath therefore needs to trace its origins to patterns of thought as well as shifts in affairs, and the two processes do not necessarily coincide. The second requirement for understanding the role of political thought in the Revolution is to reconstruct carefully the actual doctrines articulated and deployed. Irish historians have been reluctant to engage in this process of interpretation. Yet a more searching account of political ideas in the period has the potential to change our approach to the Revolution as a whole.</jats:p