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 that have subtler statements which need to keep track of universe levels rather explicitly, and are applied to get the results that require resizing.

中文翻译：

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单价数学中的内射类型

我们研究单价数学中的单射类型和代数单射类型，无论是在不存在和存在命题调整大小的情况下。内射性由限制图沿任何嵌入的超射性定义，代数内射性由限制图的给定部分沿任何嵌入定义。在命题大小调整公理下，主要结果很容易说明： (1) 内射性等价于代数内射性的命题截断。(2) 代数内射型正是宇宙指数幂的缩回。(2a) 代数内射集正是幂集的缩回。(2b) 代数内射 (n+1)-类型正是宇宙指数幂的缩回n-类型。(3) 代数内射类型也正是偏映射分类器的代数的缩回。从（2）可以看出，任何宇宙都被嵌入作为任何更大宇宙的缩回。在没有命题调整大小的情况下，我们有类似的结果，它们有更微妙的陈述，需要相当明确地跟踪宇宙水平，并应用于获得需要调整大小的结果。