We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of 0 as the image of a natural map to a product of discrete spaces. A particular case ...

Let (D, Jat) be an atomic site and j : Sh(D, Jat) → Db be the associated sheaf topos. Any functor φ : C → D induces a geometric morphism C →b Db and, by pulling-back along j, a geometric morphism q : F → Sh(D, Jat). We ...

A pre-cohesive geometric morphism p : E → S satisfies Continuity if the canonical p!(Xp ∗S) → (p!X) S is an iso for every X in E and S in S. We show that if S = Set and E is a presheaf topos then, p satisfies Continuity ...

We introduce an apparent strengthening of Sufficient Cohesion that we call Stable Connected Codiscreteness (SCC) and show that if $p: E --> S$ is cohesive and satisfies SCC then the internal axiom of choice holds in $S$. ...

Let E be a topos. If l is a level of E with monic skeleta then it makes sense to consider the objects in E that have l-skeletal boundaries. In particular, if p : E o S is a pre-cohesive geometric morphism then its centre ...

We strengthen a result of Lawvere by proving that every pre-cohesive geometric morphism p: E --> S has a canonical intensive quality s: E --> L. We also discuss examples among bounded pre-cohesive p: E --> S and, in ...

A level j : Ej → E of a topos E is said to have monic skeleta if, for every X in E, the counit j!(j∗X) → X is monic. For instance, the centre of a hyperconnected geometric morphism is such a level. We establish two related ...

Let be a topos, be the full subcategory of decidable objects, and be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity for the two subcategories of ...