Equivalence of morphisms under substitution

Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams

In a category with products, we have the diagonal morphism (WP, nLab) dup and the pairing operator pair(-, -) satisfying the point-free equivalence:

The inner pair can't be replaced with dup because the types don't line up; the two nil can't be unified. Indeed, cons ∘ dup is never well-typed, because cons takes inputs of different types but dup returns outputs of identical types. What went wrong here?

More generally, suppose we have an equivalence of polymorphic families of morphisms: For some objects $X, Y, \dots$ , we have a diagram over those objects. Each commuting pair of morphisms in that diagram should be equivalent, even if we paste it into a larger diagram. But not all equivalences type-check under substitution. Why not?

Site design / logo © 2024 Stack Exchange Inc; user contributions licensed under CC BY-SA . rev 2024.4.24.8132

Leave a Comment