# Contents

## Idea

Homotopy level (or h-level) is another name for the notion of truncation (particularly in (∞,1)-categories and their internal language of homotopy type theory) in which the numbering is offset by 2:

a homotopy n-type is a type of homotopy level $n+2$.

This offset in counting enables it to “start” at 0 rather than (-2), which is convenient when defining it by induction over the natural numbers in type theory. Thus, the correspondence between the various terminologies is indicated in the following table.

h-level 2 | 0-truncated | discrete space | 0-groupoid/set | sheaf | h-set h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | | h-2-groupoid h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | | h-3-groupoid h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | | h-$n$-groupoid | h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid

Revised on November 24, 2012 22:16:07 by Stephan Alexander Spahn (192.87.226.73)