. We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in Martin-Lof's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families. 1 Introduction In a previous paper [8] I introduced a general notion of simultaneous inductiverecursive definition in intuitionistic type theory. This notion subsumes various reflection principles and seems to pave the way for a natural development of what could be called "internal type theory", that is, the construction of models of (fragments of) type theory in type theory, and more generally, the formalization of the metatheory of type theory in type theory. The present paper is a first investigation of such an internal type theor...

this paper we describe a new, categorical approach to normalization in typed -

Datatypes which di#er inessentially in their names and structure are said to be isomorphic; for example, a ternary product is isomorphic to a nested pair of binary products. In some canonical cases, the conversion function is uniquely determined solely by the two types involved.

We present a systematic construction of a reduction-free normalization function. Starting from

In type theory a proposition is represented by a type, the type of its proofs. As a consequence the equality relation on a certain type is represented by a binary family of types. Equality on a type may be conventional or inductive. Conventional equality means that one particular equivalence relation is singled out as the equality, while inductive equality, which we also call identity, is inductively defined as the "smallest reflexive relation". It is sometimes convenient to know that the type representing a proposition is collapsed in the sense that all its inhabitants are identical. Although uniqueness of identity proofs for an arbitrary type is not derivable inside type theory, there is a large class of types for which it may be proved. Our main result is a proof that any type with decidable identity has unique identity proofs. This result is convenient for proving that the class of types with decidable identities is closed under indexed sum. Our proof of the main result...

We introduce a sequent system for basic category-theoretic reasoning suitable for computer implementation. We illustrate its use by giving a complete formal proof that the functor categories Fun[C × D, E] and Fun[C, Fun[D, E]] are naturally isomorphic.

. We compare formalizations of an example from elementary category theory in the systems HOL (an implementation of Church's classical simple type theory) and ALF (an implementation of MartinL of's intuitionistic type theory). The example is a proof of coherence for monoidal categories which was extracted from a proof of normalization for monoids. It makes essential use of the identification of proofs and programs which is fundamental to intuitionistic type theory. This aspect is naturally highlighted in the ALF formalization. However, it was possible to develop a similar formalization of the proof in HOL. The most interesting aspect of the developments concerned the implementation of diagram chasing. In particular, the HOL development was greatly facilitated by an implementation of tool support for equational reasoning in Standard ML. 1 Introduction We compare the two proof assistants ALF and HOL by using them for implementing a proof in elementary category theory 3 . This proof was...

A straightforward proof of the Coherence theorem for monoidal categories is presented in [2]. Here we describe the its formalisation within Martin-Löf type theory. The paper is organised as a commentary to the ALF proof files (recapitulated complete). In the current version the necessary categories and functors are constructed, and the main lemma (nat) is proved for all cases, except mult and inverse arrows. The case mult was already proved in the version 3a (which used different definitions incompatible with version 4)

this paper, we present a formalization of the proof in the HOL theorem prover [5], which is based on simple type theory. The formalization is considerably simpler than the ALF formalization, both theoretically and practically. Just like ALF, HOL does not directly support equational reasoning in diagram chasing. However, HOL comes with the functional programming language ML. This made it possible with a very limited programming effort to support diagram chasing at such a level that proofs in HOL were more or less as abstract as informal proofs on paper. The user does not have to worry about applying transitivity, congruence and associativity rules, only about specifying the main steps, just like in a paper-and-pencil proof. The ideas for the tool support are inspired by Paulson's higher order conversions for rewriting [11] and might be useful for other purposes than category theory. They only require the presence of a congruence (i.e. equality-like) relation as in, for instance, bisimularity proofs in program verification. Below, we first give a brief introduction to category theory in Section 2 and to the HOL system in Section 3. In Section 4 the formalization of a monoid of binary words is presented, including the normalization theorem. The formalization of a monoidal category of binary words is presented in Section 5 and in Section 6 the proof of a coherence theorem for this category is presented. Section 7 presents the tool support for diagram chasing. 2 Category Theory