Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 computer-science constructive cosmology 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 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 science 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.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 26th 2011
   • (edited Jun 26th 2011)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI have added some new material to [[Boolean algebra]] and to [[ultrafilter]]. In the former, I coined the term 'unbiased Boolean algebra' for the notion which describes Boolean algebras as equivalent to finite-product-preserving functors $Fin_+ \to Set$ from the category of finite nonempty sets, and the term $k$-biased Boolean algebra to refer to the multiplicity of ways in which Boolean algebras could be considered monadic over $Set$. In [[ultrafilter]], I added some material which gives a number of universal descriptions of the ultrafilter monad. This is in part inspired by some discussions I'm having with Tom Leinster, who remarked recently at the categories list that the ultrafilter monad could be described as a codensity monad. All this is related to the unbiased Boolean algebras and to the remarks due to Lawvere, which were described on an earlier revision; this material has been reworked.

   I have added some new material to Boolean algebra and to ultrafilter. In the former, I coined the term ’unbiased Boolean algebra’ for the notion which describes Boolean algebras as equivalent to finite-product-preserving functors $Fin_+ \to Set$ from the category of finite nonempty sets, and the term $k$-biased Boolean algebra to refer to the multiplicity of ways in which Boolean algebras could be considered monadic over $Set$.

   In ultrafilter, I added some material which gives a number of universal descriptions of the ultrafilter monad. This is in part inspired by some discussions I’m having with Tom Leinster, who remarked recently at the categories list that the ultrafilter monad could be described as a codensity monad. All this is related to the unbiased Boolean algebras and to the remarks due to Lawvere, which were described on an earlier revision; this material has been reworked.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexGreat to have you back, Todd!

   Great to have you back, Todd!

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 30th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItex(Thanks, Urs (-: ) I have recently learned through a <a href="http://mathoverflow.net/questions/69086/lawvere-theories-versus-classical-universal-algebra/69097#69097">Math Overflow discussion</a> that the usual term for what I called "$k$-biased Boolean algebra" is a $k$-valued Post algebra (after Emil Post), so I inserted this information in [[Boolean algebra]].

   (Thanks, Urs (-: ) I have recently learned through a Math Overflow discussion that the usual term for what I called “$k$-biased Boolean algebra” is a $k$-valued Post algebra (after Emil Post), so I inserted this information in Boolean algebra.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeJul 7th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexAwesome! I'm now checking how well this stuff works in constructive mathematics.

   Awesome! I’m now checking how well this stuff works in constructive mathematics.