Previously: (Weak) Factorization Systems. It’s been known since Lambek that typed lambda calculus can be modeled in a cartesian closed category, CCC. Cartesian means that you can form product…| Bartosz Milewski's Programming Cafe
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
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
