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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 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 sheaves 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.
   • CommentAuthortomr
   • CommentTimeMar 3rd 2020
   • (edited Mar 3rd 2020)
   • PermaLink
   Author: tomr
   Format: MarkdownItexI found the article https://ncatlab.org/nlab/show/field+with+one+element that explains how important the field with one element is. I am more interested in logic and type theory (and cognition, consciousness that can be modelled/implemented by them, knowledge representation and automation) and just wanted to ask this - are there some connections, analogies, applications etc. between field with one element and the (Higher order) logic, set theory, type theory (lambda calculus) or arithmetic hierarchy of (in)computable functions? Is there are such connections then they would strongly motivate me to study this field furhter. Of course, I see a lot of applications of algebraic geometry in coding and cryptography, but they are not about field with one element. Thanks. I appreciate your time, that is why only some links, key terms, list of notions suffice, all the remaining I can find myself, I just need some started, some hole through which to lean further.

   I found the article https://ncatlab.org/nlab/show/field+with+one+element that explains how important the field with one element is. I am more interested in logic and type theory (and cognition, consciousness that can be modelled/implemented by them, knowledge representation and automation) and just wanted to ask this - are there some connections, analogies, applications etc. between field with one element and the (Higher order) logic, set theory, type theory (lambda calculus) or arithmetic hierarchy of (in)computable functions? Is there are such connections then they would strongly motivate me to study this field furhter. Of course, I see a lot of applications of algebraic geometry in coding and cryptography, but they are not about field with one element. Thanks.

   I appreciate your time, that is why only some links, key terms, list of notions suffice, all the remaining I can find myself, I just need some started, some hole through which to lean further.

2. • CommentRowNumber2.
   • CommentAuthortomr
   • CommentTimeMar 3rd 2020
   • PermaLink
   Author: tomr
   Format: MarkdownItexI am starting to see - HoTT is the or The language for homotopy theories and maybe it is already expressive enough to reason about F1, or maybe HoTT should be enhanced as well?

   I am starting to see - HoTT is the or The language for homotopy theories and maybe it is already expressive enough to reason about F1, or maybe HoTT should be enhanced as well?

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 4th 2020
   • (edited Mar 4th 2020)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI guess there could be a range of $\mathbb{F}_1$ entities as in some sense most basic, along the lines of > With the identification $\mathbb{F}_1 Mod \simeq FinSet^{\ast/}$ from above it follows that the [[algebraic K-theory]] over $\mathbb{F}_1$ is _[[stable cohomotopy]]_ Is there anything to fill the gap $\mathbb{F}_1 ??? \simeq Fin \infty Grpd^{\ast/}$?

   I guess there could be a range of $\mathbb{F}_1$ entities as in some sense most basic, along the lines of

   With the identification $\mathbb{F}_1 Mod \simeq FinSet^{\ast/}$ from above it follows that the algebraic K-theory over $\mathbb{F}_1$ is stable cohomotopy

   Is there anything to fill the gap $\mathbb{F}_1 ??? \simeq Fin \infty Grpd^{\ast/}$?