An abstract view on syntax with sharing

The notion of term graph encodes a refinement of inductively generated syntax in which regard is paid to the the sharing and discard of subterms. Inductively generated syntax has an abstract expression in terms of initial algebras for certain endofunctors on the category of sets, which permits one to go beyond the set-based case, and speak of inductively generated syntax in other settings. In this paper we give a similar abstract expression to the notion of term graph. Aspects of the concrete theory are redeveloped in this setting, and applications beyond the realm of sets discussed.Comment: 26 pages; v2: final journal versio

Cultural justice and ethics: From within

Where I eventually got a lot of support, awhi, aroha, was from the PPTA when I became a member of Te Huarahi and they helped to see me through the tears and the confusion. It wasn’t because I was Maori, it was because I was a human being. I think the process I have gone through is a process of becoming human, and becoming real in thinking; yes, my experience is valid. I’m not just Maori, I’m Pakeha too. I want both of them, I am both of them. I don’t have to get up and speak fluent Maori right now. Perhaps it will come. I hope it does because I want it to, and I’m going to work on it. But I do feel that people like me, and there are many of us, are not marginal. We are bridges between cultures. This diversity has to be acknowledged, honoured and respected

Combinatorial structure of type dependency

We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of all dependent sequent calculi, and exhibiting it as the category of algebras for a monad on a presheaf category. The objects of the presheaf category encode the basic judgements of a dependent sequent calculus, while the action of the monad encodes the deduction rules; so by giving an explicit description of the monad, we obtain an explicit account of the combinatorics of type dependency. We find that this combinatorics is controlled by a particular kind of decorated ordered tree, familiar from computer science and from innocent game semantics. Furthermore, we find that the monad at issue is of a particularly well-behaved kind: it is local right adjoint in the sense of Street--Weber. In future work, we will use this fact to describe nerves for dependent type theories, and to study the coherence problem for dependent type theory using the tools of two-dimensional monad theory.Comment: 35 page