Abstract Syntax: Substitution and Binders

AbstractWe summarise Fiore et al's paper on variable substitution and binding, then axiomatise it. Generalising their use of the category F of finite sets to model untyped cartesian contexts, we let S be an arbitrary pseudo-monad on Cat and consider (S1)op: this generality includes linear contexts, affine contexts, and contexts for the Logic of Bunched Implications. Given a pseudo-distributive law of S over the (partial) pseudo-monad Tcoc−=[(−)op,Set] for free cocompletions, one can define a canonical substitution monoidal structure on the category [(S1)op,Set], generalising Fiore et al's substitution monoidal structure for cartesian contexts: this provides a natural substitution structure for the above examples. We give a concrete description of this substitution monoidal structure in full generality. We then give an axiomatic definition of a binding signature, then state and prove an initial algebra semantics theorem for binding signatures in full generality, once again extending the definitions and theorem of Fiore et al. A delicate extension of the research includes the category Pb(Injop,Set) studied by Gabbay and Pitts in their quite different analysis of binders, which we compare and contrast with that of Fiore et al

Distributive laws, pseudodistributive laws and decagons

We give alternative definitions of distributive laws and pseudodistributive laws involving the decagonal coherence conditions which naturally arise when the involved monads and pseudomonads are presented in extensive form. We then use these results to give a number of simplifications in the coherence conditions for distributive laws and pseudodistributive laws. In particular, we show that five coherence axioms suffice in the usual definition of pseudodistributive laws, we give simple descriptions of distributive laws and pseudodistributive laws in terms of (pseudo)algebra structure maps, and we give concise definitions of distributive laws and pseudodistributive laws in no-iteration form.Comment: 27 pages; added the definition of a pseudodistributive law in no-iteration for

Substitution, jumps, and algebraic effects

Contains fulltext : 129931.pdf (author's version ) (Open Access

Occupy: A People Yet To Come

The term Occupy represents a belief in the transformation of the capitalist system through a new heterogenic world of protest and activism that cannot be conceived in terms of liberal democracy, parliamentary systems, class war or vanguard politics. These conceptualisations do not articulate where power is held, nor from where transformation may issue. This collection of essays by world-leading scholars of Deleuze and Guattari examines how capitalism can be understood as a global abstract machine whose effects pervade all of life and how Occupy can be framed as a response to this as a heterogenic movement based on new tactics, revitalised democratic processes and nomadic systems of organisation. Seeing the question as a political tactic aimed at delegitimizing their protest, Occupiers refused to answer the question ‘what do you want?’, produce manifestos, elect leaders or act as a vanguard. Occupy: A People Yet to Come goes some considerable way towards providing the terms upon which this refusal can be understood within a changed landscape of political activism and the rewriting of the conventions of political protest. Including essays by Claire Colebrook, Giuseppina Mecchia, John Protevi, Rodrigo Nunes, Verena Andermatt Conley, Nicholas Thoburn, Ian Buchanan, David Burrows, Eugene Holland and Andrew Conio, the volume examines the economic predicates of capitalist economics: liberal democracy and its alternatives, the conjugation of protest and aesthetics, how occupy experiments with different types of leadership and how power, hierarchies and resistance might be understood using Deleuze and Guattari’s radical conceptualizations of debt; subjectivity, the minor and the molecular, occupation, dispersed leadership, territory, smooth space and the war machine