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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeDec 11th 2022
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexCalled for from [[state]]. I told Urs that I\'d fill this link eventually! <a href="https://ncatlab.org/nlab/revision/line+segment/1">v1</a>, <a href="https://ncatlab.org/nlab/show/line+segment">current</a>

   Called for from state. I told Urs that I'd fill this link eventually!

   v1, current

2. • CommentRowNumber2.
   • CommentAuthornLab edit announcer
   • CommentTimeDec 11th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadded distinction between synthetic geometry in material set theory and structural set theory Anonymous <a href="https://ncatlab.org/nlab/revision/diff/line+segment/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/line+segment/2">v2</a>, <a href="https://ncatlab.org/nlab/show/line+segment">current</a>

   added distinction between synthetic geometry in material set theory and structural set theory

   Anonymous

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeDec 12th 2022
   • (edited Dec 12th 2022)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexDo we really need to bring set theory into it? Ordered and unordered pairs are much simpler concepts which can be taken as primitive well before thinking about anything as complex as a foundational set theory. Also, the definition of an unordered pair in terms of ordered pairs doesn\'t work. An unordered pair is an equivalence class of ordered pairs, not an ordered pair of ordered pairs; after all, $((a,b),(b,a)) \ne ((b,a),(a,b))$ (if $a \ne b$), so this is still ordered. And at that point you may as well just take a subset, as in material set theory. The whole thing is a quagmire best avoided here, and sorted out instead on pages like [[unordered pair]] and [[ordered pair]]. ETA: And on [[pairing structure]], which I assume was written by the same Anonymous contributor; that page is very interesting!

   Do we really need to bring set theory into it? Ordered and unordered pairs are much simpler concepts which can be taken as primitive well before thinking about anything as complex as a foundational set theory.

   Also, the definition of an unordered pair in terms of ordered pairs doesn't work. An unordered pair is an equivalence class of ordered pairs, not an ordered pair of ordered pairs; after all, $((a,b),(b,a)) \ne ((b,a),(a,b))$ (if $a \ne b$), so this is still ordered. And at that point you may as well just take a subset, as in material set theory. The whole thing is a quagmire best avoided here, and sorted out instead on pages like unordered pair and ordered pair. ETA: And on pairing structure, which I assume was written by the same Anonymous contributor; that page is very interesting!

4. • CommentRowNumber4.
   • CommentAuthorMadeleine Birchfield
   • CommentTimeDec 12th 2022
   • PermaLink
   Author: Madeleine Birchfield
   Format: MarkdownItexWell, looks like said anonymous editor changed the definition of unordered pair to the correct definition involving a [[quotient set]]: https://ncatlab.org/nlab/revision/diff/line+segment/3

   Well, looks like said anonymous editor changed the definition of unordered pair to the correct definition involving a quotient set:

   https://ncatlab.org/nlab/revision/diff/line+segment/3

5. • CommentRowNumber5.
   • CommentAuthorMadeleine Birchfield
   • CommentTimeDec 12th 2022
   • PermaLink
   Author: Madeleine Birchfield
   Format: MarkdownItexToby, That same anonymous editor seems to have added a type theoretic definition of [[unordered pair]] over at that article, which probably indicates that he was thinking in type theoretic terms when writing that down, and thinking of unordered pairs as elements rather than subsets. In type theory, subsets are usually defined as an h-set with an embedding into another h-set, and in the case where the type theory uses a type judgment rather than a sequence of universes to denote types, unless one has a universe or some type of propositions, one cannot quantify over subsets as one could with elements.

   Toby,

   That same anonymous editor seems to have added a type theoretic definition of unordered pair over at that article, which probably indicates that he was thinking in type theoretic terms when writing that down, and thinking of unordered pairs as elements rather than subsets. In type theory, subsets are usually defined as an h-set with an embedding into another h-set, and in the case where the type theory uses a type judgment rather than a sequence of universes to denote types, unless one has a universe or some type of propositions, one cannot quantify over subsets as one could with elements.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeJan 3rd 2023
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexThinking type-theoretically while still writing ‘set theory’ … they sound like me! Anyway, I removed the stuff about defining pairs (at least for now), while making some minor tweaks.

   Thinking type-theoretically while still writing ‘set theory’ … they sound like me!

   Anyway, I removed the stuff about defining pairs (at least for now), while making some minor tweaks.