28 lines
636 B
Markdown
28 lines
636 B
Markdown
|
# 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
|
||
|
|
|
||
|
|
-
|