Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Galois connection between preorders
Definition
Let and be preorders (Preorder and monotone map). A Galois connection consists of monotone maps and such that, for every and ,
Under the identification of preorders with thin categories in A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps, this is exactly an adjunction. The corresponding unit and counit are the inequalities
Depends on
Used by
- A Galois connection between posets satisfies FGF=F and GFG=G Corollary
- A floor-division and multiplication adjunction between natural-number preorders Example
- Ceiling ⊣ inclusion ⊣ floor: an adjoint triple between (ℝ,≤) and (ℤ,≤) Example
- The adjoint functor theorem for ordered sets Example
- In a poset adjunction the triangle identities are automatic Proposition
- Direct and inverse image of subobjects form a Galois connection Theorem
- Direct image, preimage, and universal image form an adjoint triple on power sets Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Emily Riehl, Category Theory in Context, 2nd ed., Section 4.2 (standard reference, not scraped)
- Tom Leinster, Basic Category Theory, Example 2.2.7 (standard reference, not scraped)