19 research outputs found
Identity types and weak factorization systems in Cauchy complete categories
It has been known that categorical interpretations of dependent type theory
with Sigma- and Id-types induce weak factorization systems. When one has a weak
factorization system (L, R) on a category C in hand, it is then natural to ask
whether or not (L, R) harbors an interpretation of dependent type theory with
Sigma- and Id- (and possibly Pi-) types. Using the framework of display map
categories to phrase this question more precisely, one would ask whether or not
there exists a class D of morphisms of C such that the retract closure of D is
the class R and the pair (C, D) forms a display map category modeling Sigma-
and Id- (and possibly Pi-) types. In this paper, we show, with the hypothesis
that C is Cauchy complete, that there exists such a class D if and only if
(C,R) itself forms a display map category modeling Sigma- and Id- (and possibly
Pi-) types. Thus, we reduce the search space of our original question from a
potentially proper class to a singleton.Comment: 14 page
Type-theoretic weak factorization systems
This article presents three characterizations of the weak factorization
systems on finitely complete categories that interpret intensional dependent
type theory with Sigma-, Pi-, and Id-types. The first characterization is that
the weak factorization system (L,R) has the properties that L is stable under
pullback along R and that all maps to a terminal object are in R. We call such
weak factorization systems type-theoretic. The second is that the weak
factorization system has an Id-presentation: roughly, it is generated by
Id-types in the empty context. The third is that the weak factorization system
(L, R) is generated by a Moore relation system, a generalization of the notion
of Moore paths
Coinductive Control of Inductive Data Types
We combine the theory of inductive data types with the theory of universal measurings. By doing so, we find that many categories of algebras of endofunctors are actually enriched in the corresponding category of coalgebras of the same endofunctor. The enrichment captures all possible partial algebra homomorphisms, defined by measuring coalgebras. Thus this enriched category carries more information than the usual category of algebras which captures only total algebra homomorphisms. We specify new algebras besides the initial one using a generalization of the notion of initial algebra
Coinductive control of inductive data types
We combine the theory of inductive data types with the theory of universal
measurings. By doing so, we find that many categories of algebras of
endofunctors are actually enriched in the corresponding category of coalgebras
of the same endofunctor. The enrichment captures all possible partial algebra
homomorphisms, defined by measuring coalgebras. Thus this enriched category
carries more information than the usual category of algebras which captures
only total algebra homomorphisms. We specify new algebras besides the initial
one using a generalization of the notion of initial algebra.Comment: 21 page
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Type theoretic weak factorization systems
This thesis presents a characterization of those categories with weak factorization systems that can interpret the theory of intensional dependent type theory with Σ, Π, and identity types.
We use display map categories to serve as models of intensional dependent type theory. If a display map category (C, D) models Σ and identity types, then this structure generates a weak factorization system (L, R). Moreover, we show that if the underlying category C is Cauchy complete, then (C, R) is also a display map category modeling Σ and identity types (as well as Πtypes if (C, D) models Πtypes). Thus, our main result is to characterize display map categories (C, R) which model Σ and identity types and where R is part of a weak factorization system (L, R) on the category C. We offer three such characterizations and show that they are all equivalent when C has all finite limits. The first is that the weak factorization system (L, R) has the properties that L is stable under pullback along R and all maps to a terminal object are in R. We call such weak factorization systems type theoretic. The second is that the weak factorization system has what we call an Id-presentation: it can be built from certain categorical structure in the same way that a model of Σ and identity types generates a weak factorization system. The third is that the weak factorization system (L, R) is generated by a Moore relation system. This is a technical tool used to establish the equivalence between the first and second characterizations described.
To conclude the thesis, we describe a certain class of convenient categories of topological spaces (a generalization of compactly generated weak Hausdorff spaces). We then construct a Moore relation system within these categories (and also within the topological topos) and thus show that these form display map categories with Σ and identity types (as well as Πtypes in the topological topos)
Towards a directed homotopy type theory
In this paper, we present a directed homotopy type theory for reasoning
synthetically about (higher) categories, directed homotopy theory, and its
applications to concurrency. We specify a new `homomorphism' type former for
Martin-L\"of type theory which is roughly analogous to the identity type former
originally introduced by Martin-L\"of. The homomorphism type former is meant to
capture the notions of morphism (from the theory of categories) and directed
path (from directed homotopy theory) just as the identity type former is known
to capture the notions of isomorphism (from the theory of groupoids) and path
(from homotopy theory). Our main result is an interpretation of these
homomorphism types into Cat, the category of small categories. There, the
interpretation of each homomorphism type hom(a,b) is indeed the set of
morphisms between the objects a and b of a category C. We end the paper with an
analysis of the interpretation in Cat with which we argue that our homomorphism
types are indeed the directed version of Martin-L\"of's identity types
A Hurewicz Model Structure for Directed Topology
This paper constructs an h-model structure for diagrams of streams, locally preordered spaces. Along the way, the paper extends some classical characterizations of Hurewicz fibrations and closed Hurewicz cofibrations. The usual characterization of classical closed Hurewicz cofibrations as inclusions of neighborhood deformation retracts extends. A characterization of classical Hurewicz fibrations as algebras over a pointed Moore cocylinder endofunctor also extends. An immediate consequence is a long exact sequence for directed homotopy monoids, with applications to safety verifications for database protocols
B-systems and C-systems are equivalent
C-systems were defined by Cartmell as models of generalized algebraic
theories. B-systems were defined by Voevodsky in his quest to formulate and
prove an initiality conjecture for type theories. They play a crucial role in
Voevodsky's construction of a syntactic C-system from a term monad.
In this work, we construct an equivalence between the category of C-systems
and the category of B-systems, thus proving a conjecture by Voevodsky