Code Generation for Higher Inductive Types

Higher inductive types are inductive types that include nontrivial higher-dimensional structure, represented as identifications that are not reflexivity. While work proceeds on type theories with a computational interpretation of univalence and higher inductive types, it is convenient to encode these structures in more traditional type theories with mature implementations. However, these encodings involve a great deal of error-prone additional syntax. We present a library that uses Agda's metaprogramming facilities to automate this process, allowing higher inductive types to be specified with minimal additional syntax.Comment: 16 pages, Accepted for presentation in WFLP 201

Coalgebras on Measurable Spaces

Thesis (PhD) - Indiana University, Mathematics, 2005Given an endofunctor T in a category C, a coalgebra is a pair (X,c) consisting of an object X and a morphism c:X ->T(X). X is called the carrier and the morphism c is called the structure map of the T-coalgebra. The theory of coalgebras has been found to abstract common features of different areas like computer program semantics, modal logic, automata, non-well-founded sets, etc. Most of the work on concrete examples, however, has been limited to the category Set. The work developed in this dissertation is concerned with the category Meas of measurable spaces and measurable functions. Coalgebras of measurable spaces are of interest as a formalization of Markov Chains and can also be used to model probabilistic reasoning. We discuss some general facts related to the most interesting functor in Meas, Delta, that assigns to each measurable space, the space of all probability measures on it. We show that this functor does not preserve weak pullbacks or omega op-limits, conditions assumed in many theorems about coalgebras. The main result will be two constructions of final coalgebras for many interesting functors in Meas. The first construction (joint work with L. Moss), is based on a modal language that lets us build formulas that describe the elements of the final coalgebra. The second method makes use of a subset of the projective limit of the final sequence for the functor in question. That is, the sequence 1 <- T1 <- T 2 1 <-... obtained by iteratively applying the functor to the terminal element 1 of the category. Since these methods seem to be new, we also show how to use them in the category Set, where they provide some insight on how the structure map of the final coalgebra works. We show as an application how to construct universal Type Spaces, an object of interest in Game Theory and Economics. We also compare our method with previously existing constructions

Classifying Topoi and Preservation of Higher Order Logic by Geometric Morphisms.

Topoi are categories which have enough structure to interpret higher order logic. They admit two notions of morphism: logical morphisms which preserve all of the structure and therefore the interpretation of higher order logic, and geometric morphisms which only preserve only some of the structure and therefore only some of the interpretation of higher order logic. The question then arises: what kinds of higher order theories are preserved by geometric morphisms? It is known that certain first order theories called internal geometric theories are preserved by geometric morphisms, and these admit what are known as classifying topoi. Briefly, a classifying topos for an internal geometric theory T in a topos E is a topos E[T] such that models of T in any topos F with a geometric morphism to E are in one to one correspondence with geometric morphisms from F to E[T] over E. One useful technique for showing that a higher order theory Tau is preserved by geometric morphisms is to define an internal geometric theory T of "bad sets" for Tau and show that Tau is equivalent to the higher order theory which says "the classifying topos for T is degenerate". We set up a deduction calculus for internal geometric theories and show that it proves a contradiction if and only if the classifying topos of that theory is degenerate. We use this result to study a variant of the higher order theory of Dedekind finite objects and the higher order theory of field objects considered as ring objects with no non-trivial ideals.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/99993/1/sjhenry_1.pd