type-theory/notes/HLevels.md

# Properties

- Theorem 7.1.4:
  - IF: $X$ is an $n$ type
  - IF: $X \rightarrow Y$ is a retraction (has a left-inverse)
  - THEN: $Y$ is an $n$ type
- Corollary 7.1.5:
  - IF: $X \simeq Y$
  - IF: $X$ is an $n$ type
  - THEN: $Y$ is an $n$ type
- Theorem 7.1.7:
  - IF: $X$ is an $n$ type
  - THEN: it is also an $(n + 1)$ type
- Theorem 7.1.8:
  - IF: $A$ is an $n$ type
  - IF: $B(a)$ is an $n$ type for all $a : A$
  - THEN: $\sum_{(x : A)} B(x)$ is an $n$ type

## -2: Contractible

## -1: Mere props

- If $A$ and $B$ are mere props, so is $A \times B$

- If $B(a)$ is a prop for any $a:A$, then $\prod_{(x:A)} B(x)$ is a prop

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