Homotopy type theory is a foundation of mathematics that both expands and refines the classical foundations built on first-order logic and set theory. It accounts properly and formally for various aspects of informal mathematics that are otherwise difficult to formalize, for instance the idea that equivalent structures may be identified. It is also able to express the distinction between concrete and abstract existence arguments. It thus offers an answer, however useful it might be, to the question: what precisely is an explicit bijection?