# Simplicial Sets Notation

I just started reading about simplicial sets from Manin and Gelfand's "Methods of Homological Algebra," and I have run into some notation with which I am not familiar in their discussion of n dimensional simplexes. $I\subset [n]\stackrel{df}{=}\{1,...,n\}$

This is used to refer to the subset associated with the "I-th face of $\Delta_n$. Can someone please clarify what the "df" over the equals sign means in this context, and whether [n] in this context just refers to the set of natural numbers from $1$ to $n$?

Here $[n] \overset{df}{=} \{1,\dots,n\}$ means that $[n]$ is defined to be $\{1,\dots,n\}$. Other common notations for this are $:=$, $\overset{def}{=}$ (in fact I wouldn't be surprised if there was actually an 'e' missing), $\overset{\Delta}{=}$...
• Though, it is a bit odd to define $[n] = \{ 1, \ldots, n \}$ if one wants to work with simplices... Nov 10 '15 at 11:20
• It is more convenient to take $[n] = \{ 0, \ldots, n \}$ in that case. Nov 10 '15 at 18:55