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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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 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.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 26th 2013
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSomehow I'm reminded of that Temptations song 'Ball of confusion'. Anyway, a thread to jot down ideas on how the different varieties of cohesion relate to each other. Anyone welcome to join in. There's a question of how to organise cohesion. Do we see it as a relative notion and then gather together pairs of $\infty$-toposes, one cohesive over the other, or do we see enough from cohesion over $\infty Grpd$? We have $$ Smooth \infty Grpd \to Euclidean Top \infty Grpd \to \infty Grpd. $$ Then a cluster of infinitesimal extensions: super, synthetic, synthetic super imposed on them, except super does no work on Euclidean Top. Is there a synthetic Euclidean Top? The synthetic extension of $\infty Grpd$ is $Inf \infty Grpd$. Is there a $Super Inf \infty Grpd$? On everything we can act by $(-)^I$, and its approximations, $J^n(-)$. $J^1 = T$ on $\infty Grpd$ is infinitesimal. $\infty Grpd^I$ is not, so at some point in the interpolation this property must fail. Then there's global cohesion.

   Somehow I’m reminded of that Temptations song ’Ball of confusion’. Anyway, a thread to jot down ideas on how the different varieties of cohesion relate to each other. Anyone welcome to join in.

   There’s a question of how to organise cohesion. Do we see it as a relative notion and then gather together pairs of $\infty$-toposes, one cohesive over the other, or do we see enough from cohesion over $\infty Grpd$?

   We have

   $$Smooth \infty Grpd \to Euclidean Top \infty Grpd \to \infty Grpd.$$

   Then a cluster of infinitesimal extensions: super, synthetic, synthetic super imposed on them, except super does no work on Euclidean Top. Is there a synthetic Euclidean Top?

   The synthetic extension of $\infty Grpd$ is $Inf \infty Grpd$. Is there a $Super Inf \infty Grpd$?

   On everything we can act by $(-)^I$, and its approximations, $J^n(-)$. $J^1 = T$ on $\infty Grpd$ is infinitesimal. $\infty Grpd^I$ is not, so at some point in the interpolation this property must fail.

   Then there’s global cohesion.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 27th 2013
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   Author: David_Corfield
   Format: MarkdownItexIn section 4.6.1 of dcct, Urs represents some of the above in terms of $\{\mathbb{R}^{p \oplus s| q}\}_{p, s, q}$. There we see six versions of this with various combinations of $p$, $q$, and $s$ sent to zero. But there are eight combinations possible for three quantities. There could be $\{\mathbb{R}^{0 \oplus s| q}\}_{s, q}$ and $\{\mathbb{R}^{0 \oplus s}\}_{s}$. These would give rise to synthetic versions of the right hand pair: $Inf \infty Grpd$ and $Super Inf \infty Grpd$. This relates to that [discussion](http://nforum.mathforge.org/discussion/4973/synthetic-quantum-field-theory/?Focus=40071#Comment_40071) on Manin's three dimensions

   In section 4.6.1 of dcct, Urs represents some of the above in terms of $\{\mathbb{R}^{p \oplus s| q}\}_{p, s, q}$. There we see six versions of this with various combinations of $p$, $q$, and $s$ sent to zero. But there are eight combinations possible for three quantities. There could be $\{\mathbb{R}^{0 \oplus s| q}\}_{s, q}$ and $\{\mathbb{R}^{0 \oplus s}\}_{s}$. These would give rise to synthetic versions of the right hand pair: $Inf \infty Grpd$ and $Super Inf \infty Grpd$.

   This relates to that discussion on Manin’s three dimensions

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 5th 2013
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   Author: Urs
   Format: MarkdownItexYes! That could be and eventially should be made explicit somewhere.

   Yes! That could be and eventially should be made explicit somewhere.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 6th 2013
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   Author: David_Corfield
   Format: MarkdownItexIs there a theory about these dimensions? I mean, given $\mathbb{R}$, how do I get to $\{\mathbb{R}^{p \oplus s| q}\}_{p, s, q}$? Could I replace $\mathbb{R}$, and find similar extensions?

   Is there a theory about these dimensions? I mean, given $\mathbb{R}$, how do I get to $\{\mathbb{R}^{p \oplus s| q}\}_{p, s, q}$?

   Could I replace $\mathbb{R}$, and find similar extensions?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 6th 2013
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   Author: Urs
   Format: MarkdownItexThat the $\{\mathbb{R}^p\}_p$ themselves support a cohesive geometry rests ultimately on the fact that the Cartesian spaces are locally and globally contractible (and "supported on points", but that's so traditional a condition one hardly notices it). From this, the two extensions by even and by odd-graded infinitesimals is pretty automatic and just works. So as in earlier discussion, the real question is what other kinds of useful geometric spaces we have that are locally contractible. I still have the feeling that [[polydiscs]] should work, supporting cohesive analytic geometry. I even suspect that this is pretty obviously so. But sadly I still haven't gotten around to doing anything substantial along these lines. Somebody should.

   That the $\{\mathbb{R}^p\}_p$ themselves support a cohesive geometry rests ultimately on the fact that the Cartesian spaces are locally and globally contractible (and “supported on points”, but that’s so traditional a condition one hardly notices it). From this, the two extensions by even and by odd-graded infinitesimals is pretty automatic and just works.

   So as in earlier discussion, the real question is what other kinds of useful geometric spaces we have that are locally contractible.

   I still have the feeling that polydiscs should work, supporting cohesive analytic geometry. I even suspect that this is pretty obviously so. But sadly I still haven’t gotten around to doing anything substantial along these lines. Somebody should.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 6th 2013
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   Author: David_Corfield
   Format: MarkdownItexOK, so you'd probably get odd and even-infinitesimal extensions whichever contractible spaces were found. I've probably asked before, but does using $\{\mathbb{C}^p}_p$ rather than $\{\mathbb{R}^p}_p$ make a difference?

   OK, so you’d probably get odd and even-infinitesimal extensions whichever contractible spaces were found.

   I’ve probably asked before, but does using rather than make a difference?