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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.
   • CommentAuthorUrs
   • CommentTimeSep 3rd 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexfollowing the discussion and suggestion [in another thread](http://nforum.mathforge.org/discussion/3380/cohesive-homotopy-type-theory/?Focus=33823#Comment_33823) I have created an entry _[[motivation for cohesive toposes]]_ . This contains mainly just the result of copy-and-pasting a hasty forum-post. Feel free to edit it and tweak it.

   following the discussion and suggestion in another thread I have created an entry

   motivation for cohesive toposes .

   This contains mainly just the result of copy-and-pasting a hasty forum-post. Feel free to edit it and tweak it.

2. • CommentRowNumber2.
   • CommentAuthorjim_stasheff
   • CommentTimeSep 3rd 2012
   • PermaLink
   Author: jim_stasheff
   Format: TextIs cohesion more important for smooth as opposed to topological?

   Is cohesion more important for smooth as opposed to topological?
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 3rd 2012
   • (edited Sep 3rd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Is cohesion more important for smooth as opposed to topological? The thing is that, roughly, cohesive spaces are all _locally contractible_. The local contractions are those of the "basic cohesive droplets", in the spirit of the above discussion. So [[locally contractible topological infinity-groupoids]] are cohesive, as are [[Euclidean-topological infinity-groupoid]]. But a geometry modeled on a small full subcategory of [[Top]] that contains locally non-contractible spaces will in general not be cohesive. In particular for instance general [[topological stacks]] do not live in a cohesive $(2,1)$-topos. But [[differentiable stacks]] do, because every manifold and hence in particular every smooth manifold is locally contractible. So, I would say cohesion is as "important for" smooth geometry as it is to "locally contractible continuous geometry". But it is inapplicable to more general continuous geometry.

   Is cohesion more important for smooth as opposed to topological?

   The thing is that, roughly, cohesive spaces are all locally contractible. The local contractions are those of the “basic cohesive droplets”, in the spirit of the above discussion.

   So locally contractible topological infinity-groupoids are cohesive, as are Euclidean-topological infinity-groupoid.

   But a geometry modeled on a small full subcategory of Top that contains locally non-contractible spaces will in general not be cohesive. In particular for instance general topological stacks do not live in a cohesive $(2,1)$-topos. But differentiable stacks do, because every manifold and hence in particular every smooth manifold is locally contractible.

   So, I would say cohesion is as “important for” smooth geometry as it is to “locally contractible continuous geometry”. But it is inapplicable to more general continuous geometry.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 3rd 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have now copied the above comment [into the entry](http://ncatlab.org/nlab/show/motivation+for+cohesive+toposes#LocalContractibility), and edited a bit more.

   I have now copied the above comment into the entry, and edited a bit more.

5. • CommentRowNumber5.
   • CommentAuthorjesuslop
   • CommentTimeDec 29th 2019
   • PermaLink
   Author: jesuslop
   Format: MarkdownItexDeleted the "motivation for cohesion" related entry because it now links to this own entry. <a href="https://ncatlab.org/nlab/revision/diff/motivation+for+cohesive+toposes/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/motivation+for+cohesive+toposes/11">v11</a>, <a href="https://ncatlab.org/nlab/show/motivation+for+cohesive+toposes">current</a>

   Deleted the “motivation for cohesion” related entry because it now links to this own entry.

   diff, v11, current