Peter Freyd

Aspects of topoi

After a review of the work of Lawvere and Tierney, it is shown that every topos may be exactly embedded in a product of topoi each with 1 as a generator, and near-exactly embedded in a power of the category of sets. Several metatheorems are then derived. Natural numbers objects are shown to be characterized by exactness properties, which yield the fact that some topoi can not be exactly embedded in powers of the category of sets, indeed that the “arithmetic” arising from a topos dominates the exactness theory. Finally, several, necessarily non-elementary, conditions are shown to imply exact embedding in powers of the category of sets.

Recursive types reduced to inductive types

A setting called complete partial ordering (CPO) categories and the notion of dialgebra are described. Free dialgebras on CPO-categories are shown to be the same as minimal invariant objects. In the case that the bifunctor is independent of its contravariant variable (hence construable as a covariant functor), it is shown that minimal invariant objects serve simultaneously as initial algebras and final coalgebras. The reduction to inductive types is shown in a two-step process. First let T be a bifunctor contravariant in its first variable, convariant in the second. For each A it is possible to consider the convariant functor that sends X to TAX. If FA denotes a minimal invariant object of this covariant functor, one for each A, then F becomes a contrainvariant functor. It is shown that the minimal invariant objects of F are minimal invariant objects of the original bifunctor T. Secondly, let T be a contrainvariant functor. It is shown that the square of the functor (necessarily covariant) has the same minimal invariant objects.<<ETX>>

Splitting homotopy idempotents II

Let A be an abelian category with enough projectives and let ℋ be the quotient category of A obtained by identifying with zero all maps which factor through projectives. ℋ is the Eckmann-Hilton homotopy category.) Do idempotents split in ℋ? That is, given \( A\,\xrightarrow{e}\,A\, \in \,\mathcal{H} \), e2 ≡ e does there exist B ∈ ℋ and maps A → B, B → A such that A → B → A ≡ e, B → A → B ≡1?

Representations in Abelian Categories

Consider the general but imprecise question: given a category how nicely can it be represented in an abelian category?

Coherence theorems via knot theory

A generalization of the usual motion of symmetry for monoidal categories, called a ‘braiding’, was introduced in [3,4]. In that work, Joyal and Street showed that the free such category was the category with (geometric) braids as arrows, and gave a coherence theorem for braided monoidal categories in terms of braids. It was shown that a braiding was the appropriate notion of ‘commutativity’ for a 2-categorical version of the Eckmann-Hilton theorem (“A group object in the category of groups is an abelian group.“), to wit, “A monoid in the category of monoidal categories is a braided monoidal category”. Also in [3, 41, Joyal and Street gave an interpretation of abelian 3-cocycles in terms of braided compact closed groupoids. In [2] Freyd and Yetter showed that certain categories arising naturally from topological considerations in the work of Jones, Kauffman, Homfly, and others (esp. Kauffman [5]) are in fact braided categories satisfying a nonsymmetric generalization of compact closedness. In particular, it was shown that the category of oriented tangles modulo regular isotopy is the free braided strict pivotal category on one object generator (in the terminology of Joyal and Street [4]). This observation was then used to give a functorial view of the recently discovered knot polynomials, and to construct invariants of links, framed links and 3- manifolds. In this work, we shall use the connection between knot theory (in particular ‘formal’ knot theory in the style of Kauffman) and category theory in the opposite direction to derive coherence theorems for various generalizations of compact closed categories, both braided and (general) nonsymmetric. The authors are indebted to Andre Joyal and Ross Street for observations of errors

A Categorical Approach to Realizability and Polymorphic Types

A categorical calculus of relations is used to derive a unified setting for higher order logic and polymorphic lambda calculus.

Extensional PERs

A search is conducted for a class of PERs (partial equivalence relations on the natural numbers) such that the resulting full subcategory has the expected properties of any good category of CPOs: it should be a CCC (Cartesian closed category) and every endomorphism should have a canonical fixed point. Moreover the reflection functor (usually called the lifting operation) should yield a good notion of partial map. The following topics are discussed: conventions, partial-map classifiers, ExPERS, ExPERS as domains, reflectivity of strict maps, multicorreflectivity of strict maps, the extensional natural numbers, domain equations, and intrinsic descriptions. >

Semantic parametricity in polymorphic lambda calculus

A semantic condition necessary for the parametricity of polymorphic functions is considered. One of its instances is the stability condition for elements of variable type in the coherent domains semantics. A larger setting is presented that does not use retract pairs and keeps intact a basic feature of a certain function-type constructor. Polymorphic lambda terms are semantically parametric because of normalization.<<ETX>>

Aspects of topoi: Corrigenda and acknowledgements

“Aspects of Topoi” [1] appeared in July 1972 through the Herculean efforts of Bernard Neumann with the able assistance of Tim Brook and Max Kelly. The original manuscript was written in haste during the last two weeks of the authors visit to Australia. Somehow just five months later it appeared in print. Among the difficulties surmounted was a set of page proofs which having been sent via slow steamer by the U. of Penn. administration arrived after the paper was in press.

Logic Programming in Tau Categories

Many features of current logic programming languages are not captured by conventional semantics. Their fundamentally non-ground character, and the uniform way in which such languages have been extended to typed domains subject to constraints, suggest that a categorical treatment of constraint domains, of programming syntax and of semantics may be closer in spirit to declarative programming than conventional set theoretic semantics.