The cut functor

Construction 5.3.11. We will construct a functor \(\Cut \colon \simp \catop \to \Span (\Fin , \inj ,\all ) \subseteq \Span (\Fin )\):

• On objects, it sends \([n]\) to the set \(\lra {n} := \{1, \dots , n\}\).
• On morphisms, it sends \(\phi \colon [m] \to [n]\) to the span

  

  where the left-pointing map is the inclusion and the right-pointing map sends \(i\) to the unique \(1 \leq j \leq m\) such that \(\phi (j-1) < i \leq \phi (j)\).

More conceptually, we may think of the set \(\Cut (P)\) for a poset \(P\) as the set of ‘Dedekind cuts’ of \(P\), defined as pairs \((P_0, P_1)\) of non-empty subposets \(P_0,P_1 \subseteq P\) satisfying \(P = P_0 \cup P_1\) and \(p_0 < p_1\) for all \(p_0 \in P_0\) and \(p_1 \in P_1\). We want to think of the element \(k \in \{1, \dots , n\} = \Cut ([n])\) as the following partition of \([n]\): \[ k \quad \leftrightsquigarrow \quad (\{0, \dots , k-1\}, \{k, \dots , n\}). \] On morphisms, \(\Cut (\phi )\) is then simply given by taking preimages of the two subsets \(P_0\) and \(P_1\), which is only defined if both of these preimages are non-empty.

Alternatively, we may think of the set \(\Cut ([n])\) as labeling the \(n\) inequalities in the partially ordered set \([n]\) that we can use to cut \([n]\) into two pieces: \[ [n] \quad = \quad \{ 0 \,\, \overset {1}{\leq } \,\, 1 \,\, \overset {2}{\leq } \,\, 2 \,\, \overset {3}{\leq } \,\, \dots \,\, \overset {n}{\leq } \,\, n\}. \] This is the bookkeeping already used in Equation 5.1: the middle term of the span \(\Cut (\phi )\) consists of those inequalities of \([n]\) which lie in one of the blocks cut out by \(\phi \), the right-pointing map records for each of them the inequality of \([m]\) whose block contains it, and the left-pointing inclusion discards the inequalities of \([n]\) lying in no block at all.