Lemma 13.1.8. For integers \(0 \leq i \leq k \leq l \leq j \leq n\), the commutative square
is a pullback square in \(\Tw ^r([n])\).
Proof. Given \(0 \leq i' \leq j' \leq n\) we need to show that we have \((i',j') \leq (i,j)\) if and only if \((i',j') \leq (i,l)\) and \((i',j') \leq (k,j)\). This is clear: in both cases this amounts to asking that for both of the relations \(i' \leq i\) and \(j \leq j'\). □
Generated from the authoritative LaTeX source.