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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 definitions 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 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.
   • CommentAuthoradeelkh
   • CommentTimeDec 20th 2012
   • PermaLink
   Author: adeelkh
   Format: Markdown[infinity-category](http://ncatlab.org/nlab/show/infinity-category) I just added a link to Lurie's "What is...?" paper.

   infinity-category

   I just added a link to Lurie's "What is...?" paper.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 20th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexBut note that that article is about $(\infty, 1)$-categories.

   But note that that article is about $(\infty, 1)$-categories.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 20th 2012
   • (edited Dec 20th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks Adeel! We were indeed lacking that reference. While David is of course right about what he says above, on the other hand our old and woefully neglected entry _[[∞-category]]_ didn't really make that clear at all, nor did it really point to _[[(∞,1)-category]]_. I have now added a few paragraphs to the Idea-section at _[[∞-category]]_ that try to amplify that the term "$\infty$-category" is used to mean two rather different things, and that one of these meaning is discussed not in this entry, but in _[[(∞,1)-category]]_. I also moved the _What is..._-reference to _[[(∞,1)-category]]_.

   Thanks Adeel! We were indeed lacking that reference.

   While David is of course right about what he says above, on the other hand our old and woefully neglected entry ∞-category didn’t really make that clear at all, nor did it really point to (∞,1)-category.

   I have now added a few paragraphs to the Idea-section at ∞-category that try to amplify that the term “$\infty$-category” is used to mean two rather different things, and that one of these meaning is discussed not in this entry, but in (∞,1)-category.

   I also moved the What is…-reference to (∞,1)-category.

4. • CommentRowNumber4.
   • CommentAuthoradeelkh
   • CommentTimeDec 21st 2012
   • PermaLink
   Author: adeelkh
   Format: MarkdownNice!

   Nice!

5. • CommentRowNumber5.
   • CommentAuthornLab edit announcer
   • CommentTimeNov 3rd 2021
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexFixed the link to "Higher-Dimensional Categories: An Illustrated Guidebook" (now at Eugenia Cheng's personal website, no longer at Sheffield). Arne Hofmann <a href="https://ncatlab.org/nlab/revision/diff/infinity-category/32">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinity-category/32">v32</a>, <a href="https://ncatlab.org/nlab/show/infinity-category">current</a>

   Fixed the link to “Higher-Dimensional Categories: An Illustrated Guidebook” (now at Eugenia Cheng’s personal website, no longer at Sheffield).

   Arne Hofmann

   diff, v32, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexTo avoid this "link rot" of references I have adopted the habit of uploading to the nLab server non-stably hosted author pdf-s that I care about.

   To avoid this “link rot” of references I have adopted the habit of uploading to the nLab server non-stably hosted author pdf-s that I care about.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 31st 2024
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded the reference * Zach Goldthorpe, _Homotopy theories of $(\infty,\infty)$-categories as universal fixed points with respect to enrichment_ &lbrack;[arXiv:2307.00442](https://arxiv.org/abs/2307.00442)&rbrack; Do we still want to say the following?: > While there are several existing proposed definitions for what a single [[∞-category]] is, in the most general sense, there is no real understanding of the correct morphisms between them, hence of the correct [[(∞,1)-category]] of [[∞-categories]]. But this may of course change with time. <a href="https://ncatlab.org/nlab/revision/diff/infinity-category/34">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinity-category/34">v34</a>, <a href="https://ncatlab.org/nlab/show/infinity-category">current</a>

   Added the reference

   • Zach Goldthorpe, Homotopy theories of $(\infty,\infty)$-categories as universal fixed points with respect to enrichment [arXiv:2307.00442]

   Do we still want to say the following?:

   While there are several existing proposed definitions for what a single ∞-category is, in the most general sense, there is no real understanding of the correct morphisms between them, hence of the correct (∞,1)-category of ∞-categories. But this may of course change with time.

   diff, v34, current

8. • CommentRowNumber8.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 31st 2024
   • (edited Jan 31st 2024)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexThis article is strange. I propose to make it a disambiguation page for [[(∞,1)-category]] and [[(∞,∞)-category]], and most of the material can be moved to [[(∞,∞)-category]]. In fact, we do not have a separate article for (∞,∞)-categories, and the term redirects to this page. So perhaps this article could be renamed to [[(∞,∞)-category]], and a new disambiguation page [[∞-category]] could be created then.

   This article is strange. I propose to make it a disambiguation page for (∞,1)-category and (∞,∞)-category, and most of the material can be moved to (∞,∞)-category.

   In fact, we do not have a separate article for (∞,∞)-categories, and the term redirects to this page. So perhaps this article could be renamed to (∞,∞)-category, and a new disambiguation page ∞-category could be created then.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeFeb 2nd 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have cleared this old entry and added a few new paragraphs from scratch. This leaves much room for further expansion. <a href="https://ncatlab.org/nlab/revision/diff/infinity-category/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinity-category/35">v35</a>, <a href="https://ncatlab.org/nlab/show/infinity-category">current</a>

   I have cleared this old entry and added a few new paragraphs from scratch. This leaves much room for further expansion.

   diff, v35, current

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2024
    • PermaLink
    Author: Urs
    Format: MarkdownItexmoving all the old references on n-categories to *[[n-category]]*. <a href="https://ncatlab.org/nlab/revision/diff/infinity-category/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinity-category/35">v35</a>, <a href="https://ncatlab.org/nlab/show/infinity-category">current</a>

    moving all the old references on n-categories to n-category.

    diff, v35, current

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeFeb 16th 2024
    • (edited Feb 16th 2024)
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * Félix Loubaton, *Theory and models of $(\infty,\omega)$-categories* &lbrack;[arXiv:2307.11931](https://arxiv.org/abs/2307.11931)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/infinity-category/36">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinity-category/36">v36</a>, <a href="https://ncatlab.org/nlab/show/infinity-category">current</a>

    added pointer to:

    • Félix Loubaton, Theory and models of $(\infty,\omega)$-categories [arXiv:2307.11931]

    diff, v36, current

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2024
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * [[Zach Goldthorpe]], *Sheaves of $(\infty,\infty)$-categories* &lbrack;[arXiv:2403.069262](https://arxiv.org/abs/2403.06926)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/infinity-category/37">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinity-category/37">v37</a>, <a href="https://ncatlab.org/nlab/show/infinity-category">current</a>

    added pointer to:

    • Zach Goldthorpe, Sheaves of $(\infty,\infty)$-categories [arXiv:2403.069262]

    diff, v37, current