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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
   • CommentTimeJan 6th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWell I made a start, basing the entry on Urs's [FOM comment](https://cs.nyu.edu/pipermail/fom/2014-October/018251.html). <a href="https://ncatlab.org/nlab/revision/synthetic+homotopy+theory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/synthetic+homotopy+theory">current</a>

   Well I made a start, basing the entry on Urs’s FOM comment.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 6th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIt's probably to specific to the Blakers-Massey proof, but I have to be elsewhere.

   It’s probably to specific to the Blakers-Massey proof, but I have to be elsewhere.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 6th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItex> basing the entry on Urs's [FOM comment](https://cs.nyu.edu/pipermail/fom/2014-October/018251.html) If that's the case, we should probably reference it. <a href="https://ncatlab.org/nlab/revision/diff/synthetic+homotopy+theory/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/synthetic+homotopy+theory/2">v2</a>, <a href="https://ncatlab.org/nlab/show/synthetic+homotopy+theory">current</a>

   basing the entry on Urs’s FOM comment

   If that’s the case, we should probably reference it.

   diff, v2, current

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 6th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSorry. Yes, that's certainly true.

   Sorry. Yes, that’s certainly true.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 6th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded proposed advantages of cubical type theory. <a href="https://ncatlab.org/nlab/revision/diff/synthetic+homotopy+theory/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/synthetic+homotopy+theory/3">v3</a>, <a href="https://ncatlab.org/nlab/show/synthetic+homotopy+theory">current</a>

   Added proposed advantages of cubical type theory.

   diff, v3, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added this pointer: * [[Dan Licata]]: _Homotopy theory in type theory_, 2013 ([pdf slides](http://dlicata.web.wesleyan.edu/pubs/bll13homotopy/bll13homotopy.pdf), [[Licata-HomotopyInTypeTheory.pdf:file]], [blog entry 1](https://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-type-theory-introduction), [blog entry 2](https://homotopytypetheory.org/2013/05/20/homotopy-theory-in-type-theory-progress-report/)) Each time I am looking for where to link to for exposition of the subject, I find these slides, form almost 10 years ago, are still among the best available. <a href="https://ncatlab.org/nlab/revision/diff/synthetic+homotopy+theory/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/synthetic+homotopy+theory/5">v5</a>, <a href="https://ncatlab.org/nlab/show/synthetic+homotopy+theory">current</a>

   I have added this pointer:

   • Dan Licata: Homotopy theory in type theory, 2013 (pdf slides, Licata-HomotopyInTypeTheory.pdf:file, blog entry 1, blog entry 2)

   Each time I am looking for where to link to for exposition of the subject, I find these slides, form almost 10 years ago, are still among the best available.

   diff, v5, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded cross-link with *[[Awodey's conjecture]]* and pointer to * [[Steve Awodey]], §3 of: *Type theory and homotopy*, in: Dybjer P., Lindström S., Palmgren E., Sundholm G. (eds.), _Epistemology versus Ontology_, Springer (2012) 183-201 $[$[arXiv:1010.1810](http://arxiv.org/abs/1010.1810), [doi:10.1007/978-94-007-4435-6_9](https://doi.org/10.1007/978-94-007-4435-6_9)$]$ <a href="https://ncatlab.org/nlab/revision/diff/synthetic+homotopy+theory/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/synthetic+homotopy+theory/6">v6</a>, <a href="https://ncatlab.org/nlab/show/synthetic+homotopy+theory">current</a>

   added cross-link with Awodey’s conjecture and pointer to

   • Steve Awodey, §3 of: Type theory and homotopy, in: Dybjer P., Lindström S., Palmgren E., Sundholm G. (eds.), Epistemology versus Ontology, Springer (2012) 183-201 $[$arXiv:1010.1810, doi:10.1007/978-94-007-4435-6_9$]$

   diff, v6, current