Cubical Sets as a Classifying Topos∗

Coquand’s cubical set model for homotopy type theory provides the basis for a computational interpretation of the univalence axiom and some higher inductive types, as implemented in the cubical proof assistant. We show that the underlying cube category is the opposite of the Lawvere theory of De Morgan algebras. The topos of cubical sets itself classifies the theory of ‘free De Morgan algebras’. This provides us with a topos with an internal ‘interval’. Using this interval we construct a model of type theory following van den Berg and Garner. We are currently investigating the precise relation with Coquand’s. We do not exclude that the interval can also be used to construct other models. The topos of cubical sets Simplicial sets from a standard framework for homotopy theory. The topos of simplicial sets is the classifying topos of the theory of strict linear orders with endpoints. Cubical sets turn out to be more amenable to a constructive treatment of homotopy type theory. Grandis and Mauri [4] describe the classifying theories for several cubical sets without diagonals. We consider the most recent cubical set model [2]. This consists of symmetric cubical sets with connections (∧,∨), reversions (¬) and diagonals. Let F be the category of finite sets with all maps. Consider the monad DM on F which assigns to each finite set F the finite free DM-algebra (=De Morganalgebra) on F . That this set is finite can be seen using the disjunctive normal form. The cube category in [2] is the Kleisli category for the monad DM . Lawvere theory Recall that for each algebraic (=finite product) theory T , the Lawvere theory Cfp[T ] is the opposite of the category of free finitely generated models. This is the classifying category for T : models of T in any finite product category category E correspond to product-preserving functors Cfp[T ] → E. The Kleisli category KLDM is precisely the opposite of the Lawvere theory for DM-algebras: maps I → DM(J) are equivalent to DMmaps DM(I)→ DM(J) since each such DM-map is completely determined by its behavior on the atoms, as DM(I) is free. Classifying topos To obtain the classifying topos for an algebraic theory, we first need to complete with finite limits, i.e. to consider the category Cfl as the opposite of finitely presented DM-algebras. Then C fl → Set, i.e. functors on finitely presented T -algebras, is the classifying topos. This topos contains a generic T -algebra M . T -algebras in any topos F correspond to left exact left adjoint functors from the classifying topos to F . Let FG be the category of free finitely generated DM-algebras and let FP the category of finitely presented ones. We have a fully faithful functor f : FG→ FP . This gives a geometric morphism φ between the functor toposes. Since f is fully faithful, φ is an embedding. ∗I gratefully acknowledge the support of the Air Force Office of Scientific Research through MURI grant FA9550-15-1-0053. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the AFOSR.