Sectional category of maps related to finite spaces (Q6614459)

Let \(f\colon E\to B\) a continuous map. The sectional category \(\mathrm{secat}(f)\) is the smallest number \(n\) such that \(B\) is covered by \(n + 1\) open sets \(U_0, \dots , U_n\) where each \(U_i\) admits a local homotopy section of \(f\).\N\NIn this paper, the author compute some examples of the sectional category for continuous maps related to finite spaces. As finite spaces are not Hausdorff in general, the sectional category in finite spaces behaves differently from that in Hausdorff spaces. For instance, the sectional category of the weak homotopy equivalence \(\tau_P\colon \mathrm{sd}(P) \to P\) from the barycentric subdivision \(\mathrm{sd}(P)\) of a finite space \(P\) to \(P\) is not zero. The author shows on the contrary that \(\mathrm{secat}(\tau_P) = \mathrm{cat}(P)\) for a finite space \(P\) with \(\mathrm{cat}(P) \leq 1\) or height 1. He also shows that there exists a finite space \(P\) satisfying \(\mathrm{secat}(\tau_P) = 1\) and \(\mathrm{cat}(P) = 2\).\N\NThen the author introduces the definition of \(k\)-th sectional category \(\mathrm{secat}_k (f)\) of a map \(f\colon P \to Q\) between finite spaces. Consider the following inverse system of barycentric subdivisions: \[Q \overset{\tau_1}{\longleftarrow} \mathrm{sd}(Q) \overset{\tau_2}{\longleftarrow} \mathrm{sd}^2(Q) \overset{\tau_3}{\longleftarrow} \mathrm{sd}^3(Q) \overset{\tau_4}{\longleftarrow} \dots\] The author defines \(\mathrm{secat}_k (f)\) as the smallest number \(n\) such that there exist \(n+1\) open sets \(U_0, \dots , U_n\) covering \(\mathrm{sd}^k(Q)\) where each \(U_i\) admits a map \(s_i\colon U_i \to X\) satisfying \(f \circ s_i \simeq \tau_1 \circ \dots \circ \tau_k|_{U_i}\).\N\NFor any continuous map \(f \colon X \to Y\) between finite (realized) simplicial complexes \(X\) and \(Y\), the simplicial approximation theorem provides a finite model of \(f\), that is, a map \(f' \colon P \to Q\) between finite spaces \(P\) and \(Q\) such that the associated map \(\mathcal{B}(f')\) on the classifying spaces is homotopic to \(f\). Note that the inequality \(\mathrm{secat}(f) \leq \mathrm{secat}(f')\) always holds and the gap between them can be arbitrarily large. The author shows that the equality \(\mathrm{secat}(f) = \mathrm{secat}_k(f')\) holds for sufficiently large \(k\).