We introduce a category that represents varying risk as well as ambiguity. We give a generalized conditional expectation as a presheaf for this category, which not only works as a traditional conditional expectation given a σ-field but also is compatible with change of measure. Then, we reformulate dynamic monetary value measures as a presheaf for the category. We show how some axioms of dynami...

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireeectivity. Bireeective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterize them in terms of split-idempotent natural transformations on id A. In the special case that A is a presheaf category, we characteri...

Name passing calculi are nowadays an established field on its own. Besides their practical relevance, they offered an intriguing challenge, since the standard operational, denotational and logical methods often proved inadequate to reason about these formalisms. A domain which has been successfully employed for languages with asymmetric communication, like the π-calculus, are presheaf categorie...

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of preshe...

ion and application , x : P ⊢ t : Q ⊢ x.t : P ⊸ Q P t −→ Q curry t −−−−→ P ⊸ Q ⊢ t : P ⊸ Q ⊢ u : P , ⊢ t · u : Q t −→ P ⊸ Q u −→ P (uncurry t) ◦ (1 u) −−−−−−−−−−−−−→ Q Recursion and sum , x : P ⊢ t : P ⊢ rec x.t : P P t −→ P fix t −−−→ P ⊢ ti : P all i ∈ I ⊢ Σi∈Iti : P ti −→ P all i ∈ I Σi∈Iti −−−−→ P In the rule for recursion, x, y must not be crossed in t, for any y in .

Some applications are displayed: these include a Galois cohomological descent spectral sequence for the étale K–theory of a scheme (where the Galois group is the Grothendieck fundamental group), and the Morel–Voevodsky description of Thomason–Trobaugh K–theory as Nisnevich K–theory in nonnegative degrees. The is also a spectrum-level description of Voevodsky’s periodicity operator for Nisnevich...

We propose a new type theory with internalized parametricity. Compared to previous similar proposals, this version comes with a denotational semantics which is a refinement of the standard presheaf semantics of dependent type theory. Further, this presheaf semantics is a refinement of the one used to interpret nominal sets with restriction. The present calculus is a candidate for the core of a ...

Higher-order recursion schemes are recursive equations defining new operations from given ones called “terminals”. Every such recursion scheme is proved to have a least interpreted semantics in every Scott’s model of λ-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite λ-terms we follow the idea of Fiore, Plotkin and Turi ...

The category of sheaves over a topological space has a rich internal logic, but they also have a very rich geometric interpretation. In these notes I review the basic notions of sheaves over topological spaces, with an emphasis on the geometry of sheaves. I prove that sheaves are toposes, present the proof of the equivalence between the category of sheaves over X and the set of local homeomorph...

In [Sco] D. Scott has shown how the interpretation of intuitionistic set theory IZF in presheaf toposes can be reformulated in a more concrete fashion à la forcing as known to set theorists. In this note we show how this can be adapted to the more general case of Grothendieck toposes dealt with abstractly in [Fou, Hay].