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The five-sequence of adjoints for combinatorial simplicial complexes

Gunnar Fløystad

Abstract

For a set $A$ let ${\mathbf {SC}_A}$ be the poset of simplicial complexes whose vertices are in $A$. For a function $f : A \rightarrow B$ there are functors $ f^{! !}, f^{**}, f^{ii}: {\mathbf {SC}_A} \rightarrow {\mathbf {SC}_B}, \quad f^{!*}, f^{i*} : {\mathbf {SC}_B} \rightarrow {\mathbf {SC}_A}, $ forming a five sequence of adjoints $f^{ !!} \dashv f^{* !} \dashv f^{* *} \dashv f^{*i} \dashv f^{ii}$. We investigate in detail these functors, and use this to give three categorical structures on simplicial complexes on finite sets such that the Stanley-Reisner correspondence to commutative monomial rings gives dualities.

The five-sequence of adjoints for combinatorial simplicial complexes

Abstract

For a set let be the poset of simplicial complexes whose vertices are in . For a function there are functors forming a five sequence of adjoints . We investigate in detail these functors, and use this to give three categorical structures on simplicial complexes on finite sets such that the Stanley-Reisner correspondence to commutative monomial rings gives dualities.
Paper Structure (27 sections, 26 theorems, 52 equations)

This paper contains 27 sections, 26 theorems, 52 equations.

Table of Contents

  1. Preliminaries on posets
  2. Galois correspondences and minimal and maximal elements
  3. The lattice of cuts in a posets
  4. The general setting
  5. Simplicial complexes and the five-sequence of adjoints
  6. Simplicial complexes
  7. The five-sequence of adjoints
  8. General statement and proofs
  9. Functoriality of Stanley-Reisner correspondence
  10. Stanley-Reisner ring
  11. Categories of simplicial complexes
  12. Categories of rings
  13. Functors from simplicial complexes to commutative rings
  14. Alexander duality
  15. Alexander duality for posets
  16. ...and 12 more sections

Key Result

Lemma 1.1

Given such a Galois correspondence. Let $p_0 \in P$ and $q_0 \in Q$.

Theorems & Definitions (59)

  • Lemma 1.1
  • Example 1.2
  • Corollary 1.3
  • proof
  • Proposition 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • Lemma 3.1
  • ...and 49 more