# Injective types in univalent mathematics

## TL;DR

This paper explores injective and algebraically injective types within univalent mathematics, establishing their characterizations and relationships to universe structures, with results depending on the presence or absence of propositional resizing axioms.

## Contribution

It provides a comprehensive analysis of injective types in univalent foundations, linking them to propositional truncation, universe retracts, and partial-map classifiers, with new characterizations under different axiomatic assumptions.

## Key findings

- Injectivity equals propositional truncation of algebraic injectivity under resizing.
- Algebraically injective types are retracts of exponential powers of universes.
- Results extend to algebraically injective sets and higher types, with universe embedding properties.

## Abstract

We investigate the injective types and the algebraically injective types in univalent mathematics, both in the absence and in the presence of propositional resizing. Injectivity is defined by the surjectivity of the restriction map along any embedding, and algebraic injectivity is defined by a given section of the restriction map along any embedding. Under propositional resizing axioms, the main results are easy to state: (1) Injectivity is equivalent to the propositional truncation of algebraic injectivity. (2) The algebraically injective types are precisely the retracts of exponential powers of universes. (2a) The algebraically injective sets are precisely the retracts of powersets. (2b) The algebraically injective $(n+1)$-types are precisely the retracts of exponential powers of universes of $n$-types. (3) The algebraically injective types are also precisely the retracts of algebras of the partial-map classifier. From (2) it follows that any universe is embedded as a retract of any larger universe. In the absence of propositional resizing, we have similar results which have subtler statements that need to keep track of universe levels rather explicitly, and are applied to get the results that require resizing.