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nLab > Latest Changes: Kan object

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- CommentRowNumber1.
- CommentAuthorStephan A Spahn
- CommentTimeJan 9th 2012
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Author: Stephan A Spahn
Format: MarkdownItexI created a seperate [[Kan object]] which was desribed in [[internal infinity-groupoid]] before. The combinatoric part in the motivation is not needed, yet.

I created a seperate Kan object which was desribed in internal infinity-groupoid before. The combinatoric part in the motivation is not needed, yet.
- CommentRowNumber2.
- CommentAuthorMike Shulman
- CommentTimeJan 10th 2012
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Author: Mike Shulman
Format: MarkdownItexWhat does "where sSet is regarded as a Set-enriched category" mean? It hardly seems necessary... Also, I don't understand the third equivalent condition for X to be Kan. What is $\Delta^k[n]$? Can you spell it out more? I didn't know there was any way to define a Kan complex in terms of some map being a bijection.

What does “where sSet is regarded as a Set-enriched category” mean? It hardly seems necessary…

Also, I don’t understand the third equivalent condition for X to be Kan. What is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta^k[n]</annotation></semantics></math>$ ? Can you spell it out more? I didn’t know there was any way to define a Kan complex in terms of some map being a bijection.
- CommentRowNumber3.
- CommentAuthorStephan A Spahn
- CommentTimeJan 10th 2012
- (edited Jan 10th 2012)
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Author: Stephan A Spahn
Format: MarkdownItex> What does "where sSet is regarded as a Set-enriched category" mean? It hardly seems necessary... Yes, this is not necessary. I wrote it this way to motivate the generalization. > Also, I don't understand the third equivalent condition for X to be Kan. What is $\Delta^k[n]$? Can you spell it out more? I didn't know there was any way to define a Kan complex in terms of some map being a bijection. I' m sorry, there I did not type what I meant: the correct condition would be ''$[\Delta[n],X]\xrightarrow{d} K_n:=\{(x_0,...,x_{k-1},x_{k+1},...,x_n|d_i x_j =d_{j-1} x_i, i\neg = k, j\neg =k, i\lt j\}$ is an epimorphism ( $[\Lambda^k[n],X]$ is then in bijection with $K_n$ where $d_i$ denotes the i-th face map and $d:=d_0\times...\times d_{k_1}\times d_{k+1}\times...\times d_n$). '' So this condition is just a combinatoric description of $[\Lambda^k_n,X]$. The intention behind this is to think of the horn as a collection of n consecutive (n-1)-simplices which the Kan condition then closes by adding a further (n-1)-cell. I deleted this condition since it is not used in the generalization. Maybe one could add some hints to conditions making a [[Kan objecct]] an [[internal infinity-groupoid|internal ∞-groupoid]]. The nlab is down at the moment, so I can't look up what there already stands in regard to this. Maybe one can use some description of objects of composable n-cells -someting like $\times^{0\le i\le n}_{s,t}[\Delta[n],X]$ where $s:=d_n\star d_{n-1}\star d_{n-4}\star...$ and $t:=d_{n-1}\star_{n-3}\star d_{n-5}$ are the pasting of degeneracies- too. (Obviously something is wrong with the formatting)

What does “where sSet is regarded as a Set-enriched category” mean? It hardly seems necessary…

Yes, this is not necessary. I wrote it this way to motivate the generalization.

Also, I don’t understand the third equivalent condition for X to be Kan. What is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta^k[n]</annotation></semantics></math>$ ? Can you spell it out more? I didn’t know there was any way to define a Kan complex in terms of some map being a bijection.

I’ m sorry, there I did not type what I meant: the correct condition would be

” $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mi>d</mi></mover><msub><mi>K</mi> <mi>n</mi></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>x</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">|</mo><msub><mi>d</mi> <mi>i</mi></msub><msub><mi>x</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>d</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>x</mi> <mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>¬</mo><mo>=</mo><mi>k</mi><mo>,</mo><mi>j</mi><mo>¬</mo><mo>=</mo><mi>k</mi><mo>,</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[\Delta[n],X]\xrightarrow{d} K_n:=\{(x_0,...,x_{k-1},x_{k+1},...,x_n|d_i x_j =d_{j-1} x_i, i\neg = k, j\neg =k, i\lt j\}</annotation></semantics></math>$ is an epimorphism ( $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Lambda^k[n],X]</annotation></semantics></math>$ is then in bijection with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math>$ where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i</annotation></semantics></math>$ denotes the i-th face map and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>:</mo><mo>=</mo><msub><mi>d</mi> <mn>0</mn></msub><mo>×</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>×</mo><msub><mi>d</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msub><mo>×</mo><msub><mi>d</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>×</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>×</mo><msub><mi>d</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">d:=d_0\times...\times d_{k_1}\times d_{k+1}\times...\times d_n</annotation></semantics></math>$ ). ”

So this condition is just a combinatoric description of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>Λ</mi> <mi>n</mi> <mi>k</mi></msubsup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Lambda^k_n,X]</annotation></semantics></math>$ . The intention behind this is to think of the horn as a collection of n consecutive (n-1)-simplices which the Kan condition then closes by adding a further (n-1)-cell.

I deleted this condition since it is not used in the generalization.

Maybe one could add some hints to conditions making a Kan objecct an internal ∞-groupoid. The nlab is down at the moment, so I can’t look up what there already stands in regard to this. Maybe one can use some description of objects of composable n-cells -someting like $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>×</mo> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow> <mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></msubsup><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\times^{0\le i\le n}_{s,t}[\Delta[n],X]</annotation></semantics></math>$ where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>:</mo><mo>=</mo><msub><mi>d</mi> <mi>n</mi></msub><mo>⋆</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⋆</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>4</mn></mrow></msub><mo>⋆</mo><mo>.</mo><mo>.</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">s:=d_n\star d_{n-1}\star d_{n-4}\star...</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>:</mo><mo>=</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mo>⋆</mo> <mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub><mo>⋆</mo><msub><mi>d</mi> <mrow><mi>n</mi><mo>−</mo><mn>5</mn></mrow></msub></mrow><annotation encoding="application/x-tex">t:=d_{n-1}\star_{n-3}\star d_{n-5}</annotation></semantics></math>$ are the pasting of degeneracies- too.

(Obviously something is wrong with the formatting)
- CommentRowNumber4.
- CommentAuthorMike Shulman
- CommentTimeJan 10th 2012
- PermaLink
Author: Mike Shulman
Format: MarkdownItexOkay, I see; thanks!

Okay, I see; thanks!
- CommentRowNumber5.
- CommentAuthorDmitri Pavlov
- CommentTimeJan 29th 2021
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Author: Dmitri Pavlov
Format: MarkdownItexAdded examples. <a href="https://ncatlab.org/nlab/revision/diff/Kan+object/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Kan+object/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Kan+object">current</a>

Added examples.

diff, v7, current

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nLab > Latest Changes: Kan object