In this blog post, I’ll reformulate Ingo Blechschmidt’s “synthetic quasi-coherence” axiom — or more precisely his “general nullstellensatz” — as a lifting property inspired by Ivan Di Liberti’s work on coherent toposes and ultrastructures. This lifting property shows that synthetic quasi-coherence can be derived from a sort of “directed path induction” for toposes, suggesting the possibility of an internal logic for all toposes in which the axioms for any sort of synthet...| Topos Institute
[Ed.] C.B. has provided a much better typeset PDF version of this blog post. Dependent types are ubiquitous in mathematics, pure and applied. When we say “let be a vector of length ,” we make the collection of values to which may belong dependent upon the value of . Such dependency of types-of-things on values-of-things is fundamental to our ability to express complex mathematical ideas and build up sophisticated abstractions. By taking this essential idea to heart, dependent type theory ...| Topos Institute
One of our central goals here at Topos is to enable a variety of different people with different domains of expertise and interest to collaborate on the design and analysis of conceptually defined models. For people to collaborate on the modelling process, we’ll need a version control system for models. A core tenet of our approach to modelling is that models — both the concepts involved and any measured data — should themselves be structured data and not just the code that describes ho...| Topos Institute
Type substitution (or Liskov Substitution Principle in object-oriented contexts) allows an object of a certain type (supertype, superclass) to be replaced with an object of another type (subtype, subclass). Let’s clarify the difference between subclass and subtype (and their super- counterparts) first. They are closely related, but stem from different concepts: inheritance and type compatibility. Subtype vs. Subclass Subclass is a class that inherits from another class, known as its supercl...| Oleksandr Manenko's Blog
At TYPES 2023 I had the honor of giving an invited talk “On Isomorphism Invariance and Isomorphism Reflection in Type Theory” in which I discussed isomorphism reflection, which states that isomorphic types are judgementally equal. This strange principle is consistent, and it validates some fairly strange type-theoretic statements.| math.andrej.com