Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 23rd 2013
   • (edited Jan 23rd 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have briefly recorded the equivalence of [[FinSet]]${}^{op}$ with finite Booplean algebras at _[FinSet -- Properties -- Opposite category](http://ncatlab.org/nlab/show/FinSet#OppositeCategory)_. Then I linked to this from various related entries, such as _[[finite set]]_, _[[power set]]_, _[[Stone duality]]_, _[[opposite category]]_. (I thought we long had that information on the $n$Lab, but it seems we didn't)

   I have briefly recorded the equivalence of FinSet${}^{op}$ with finite Booplean algebras at FinSet – Properties – Opposite category. Then I linked to this from various related entries, such as finite set, power set, Stone duality, opposite category.

   (I thought we long had that information on the $n$Lab, but it seems we didn’t)

2. • CommentRowNumber2.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeApr 11th 2014
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexAdded to _[[FinSet]]_ a remark on the opposite category $FinSet^{op}$ from a constructive perspective: "In constructive mathematics, for any flavor of _[[finite set|finite]]_, $\mathcal{P}$ defines an equivalence of $FinSet$ with the opposite category of that of those [[complete lattice|complete]] [[atom|atomic]] [[Heyting algebras]] whose set of atomic elements is finite (in the same sense as in the definition of $FinSet$)." I don't know whether for some values of _finite_, this characterization can be made more interesting, i.e. whether we can give a condition which does not explicitly mention the set of atomic elements.

   Added to FinSet a remark on the opposite category $FinSet^{op}$ from a constructive perspective:

   “In constructive mathematics, for any flavor of finite, $\mathcal{P}$ defines an equivalence of $FinSet$ with the opposite category of that of those complete atomic Heyting algebras whose set of atomic elements is finite (in the same sense as in the definition of $FinSet$).”

   I don’t know whether for some values of finite, this characterization can be made more interesting, i.e. whether we can give a condition which does not explicitly mention the set of atomic elements.

3. • CommentRowNumber3.
   • CommentAuthorJohn Baez
   • CommentTimeOct 22nd 2020
   • PermaLink
   Author: John Baez
   Format: MarkdownItexAdded facts about the universal properties of FinSet and its opposite. <a href="https://ncatlab.org/nlab/revision/diff/FinSet/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/FinSet/17">v17</a>, <a href="https://ncatlab.org/nlab/show/FinSet">current</a>

   Added facts about the universal properties of FinSet and its opposite.

   diff, v17, current

4. • CommentRowNumber4.
   • CommentAuthorHurkyl
   • CommentTimeOct 22nd 2020
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexTypo fix: $FinSet^op$ is freely generated by finite limits (not finite colimits). <a href="https://ncatlab.org/nlab/revision/diff/FinSet/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/FinSet/18">v18</a>, <a href="https://ncatlab.org/nlab/show/FinSet">current</a>

   Typo fix: $FinSet^op$ is freely generated by finite limits (not finite colimits).

   diff, v18, current