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Atrium > Mathematics, Physics & Philosophy: Possibly dumb but likely easy question about natural transformations inducing a functor

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- CommentRowNumber1.
- CommentAuthorHarry Gindi
- CommentTimeJun 3rd 2010
- PermaLink
Author: Harry Gindi
Format: MarkdownItexLet $F,G\in [C,D]$ be two functors commuting with small colimits, and let $\alpha: F \to G$ be a natural transformation between them. Then according to Cisinski's book [Astérisque 308](http://www-math.univ-paris13.fr/~cisinski/ast.pdf) Corollaire 1.1.8, the natural transformation $\alpha$ induces a functor $H: D\to Arr(D)$ that commutes with small colimits as well. That's it. That's the only information he gives about $H$. I assume that it's trivial because he doesn't go into more detail, but I have no idea what it should be. If I had to guess, and this is _very_ speculative, I would say that for $X$ in $D$, we can look at the diagram induced by $\alpha$, that is, the diagram defined by $\{\alpha_Z:X\to G(Z) | Z\in F^{-1}(X)\}$ and take $$H(X):=X\to colim_{Z\in F^{-1}(X)}G(Z),$$ the induced map to the colimit.

Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F,G\in [C,D]</annotation></semantics></math>$ be two functors commuting with small colimits, and let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>:</mo><mi>F</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\alpha: F \to G</annotation></semantics></math>$ be a natural transformation between them.

Then according to Cisinski’s book Astérisque 308 Corollaire 1.1.8, the natural transformation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>$ induces a functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>Arr</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H: D\to Arr(D)</annotation></semantics></math>$ that commutes with small colimits as well. That’s it. That’s the only information he gives about $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>$ . I assume that it’s trivial because he doesn’t go into more detail, but I have no idea what it should be.

If I had to guess, and this is very speculative, I would say that for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>$ , we can look at the diagram induced by $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>$ , that is, the diagram defined by $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>α</mi> <mi>Z</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>Z</mi><mo>∈</mo><msup><mi>F</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\alpha_Z:X\to G(Z) | Z\in F^{-1}(X)\}</annotation></semantics></math>$ and take
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>X</mi><mo>→</mo><msub><mi>colim</mi> <mrow><mi>Z</mi><mo>∈</mo><msup><mi>F</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mi>G</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">H(X):=X\to colim_{Z\in F^{-1}(X)}G(Z),</annotation></semantics></math>$
the induced map to the colimit.
- CommentRowNumber2.
- CommentAuthorDavidRoberts
- CommentTimeJun 3rd 2010
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Author: DavidRoberts
Format: MarkdownItexThe natural transformation induces a functor $C' \to Arr(C)$ (switching to his notation, C replaced by C' and D replaced by C) - this is standard and trivial. I wonder if it is a typo?... It is a typo, because by the following paragraph, he refers to $F' = H^{-1}(\tilde{F})$, using the notation of the corollary this is a class of arrows in $C'$, hence $H$ is a functor from $C'$ to $Arr(C)$. The only potentially tricky bit is showing that $H$ commutes with inductive limits. I leave it as an exercise.

The natural transformation induces a functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>′</mo><mo>→</mo><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C' \to Arr(C)</annotation></semantics></math>$ (switching to his notation, C replaced by C’ and D replaced by C) - this is standard and trivial. I wonder if it is a typo?… It is a typo, because by the following paragraph, he refers to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>′</mo><mo>=</mo><msup><mi>H</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F' = H^{-1}(\tilde{F})</annotation></semantics></math>$ , using the notation of the corollary this is a class of arrows in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math>$ , hence $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>$ is a functor from $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(C)</annotation></semantics></math>$ . The only potentially tricky bit is showing that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>$ commutes with inductive limits. I leave it as an exercise.
- CommentRowNumber3.
- CommentAuthorHarry Gindi
- CommentTimeJun 3rd 2010
- PermaLink
Author: Harry Gindi
Format: MarkdownItexAh, so it was a typo!

Ah, so it was a typo!
- CommentRowNumber4.
- CommentAuthorHarry Gindi
- CommentTimeJun 3rd 2010
- PermaLink
Author: Harry Gindi
Format: MarkdownItexHmm... Then in Proposition 1.1.7, how is $f\mapsto \tilde{f}$ an endofunctor of $Arr(C)$? It's sending arrows in $Arr(C)$ to objects in $Arr(C)$. Rather, it seems like a functor $Arr(Arr(C))\to Arr(C)$, which is not an endofunctor. Is this also a mistake, or am I missing the point...

Hmm… Then in Proposition 1.1.7, how is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>↦</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">f\mapsto \tilde{f}</annotation></semantics></math>$ an endofunctor of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(C)</annotation></semantics></math>$ ? It’s sending arrows in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(C)</annotation></semantics></math>$ to objects in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(C)</annotation></semantics></math>$ . Rather, it seems like a functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>Arr</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(Arr(C))\to Arr(C)</annotation></semantics></math>$ , which is not an endofunctor. Is this also a mistake, or am I missing the point…
- CommentRowNumber5.
- CommentAuthorTodd_Trimble
- CommentTimeJun 3rd 2010
- (edited Jun 3rd 2010)
- PermaLink
Author: Todd_Trimble
Format: MarkdownItex(Comment deleted.)

(Comment deleted.)
- CommentRowNumber6.
- CommentAuthorHarry Gindi
- CommentTimeJun 3rd 2010
- PermaLink
Author: Harry Gindi
Format: MarkdownItexAlright, I've concluded that calling it an endofunctor is incorrect. If it is a functor Arr(Arr(C))->Arr(C), this makes sense with the proof, since it turns a retract of squares into a retract of arrows, and it preserves identities blah blah.

Alright, I’ve concluded that calling it an endofunctor is incorrect. If it is a functor Arr(Arr(C))->Arr(C), this makes sense with the proof, since it turns a retract of squares into a retract of arrows, and it preserves identities blah blah.
- CommentRowNumber7.
- CommentAuthorDavidRoberts
- CommentTimeJun 3rd 2010
- PermaLink
Author: DavidRoberts
Format: MarkdownItexHmm... indeed!

Hmm… indeed!
- CommentRowNumber8.
- CommentAuthorHarry Gindi
- CommentTimeJun 3rd 2010
- (edited Jun 3rd 2010)
- PermaLink
Author: Harry Gindi
Format: MarkdownItexAlright, I e-mailed Cisinski about the errors. I'm probably the first person who didn't skip through this section.

Alright, I e-mailed Cisinski about the errors. I’m probably the first person who didn’t skip through this section.

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Atrium > Mathematics, Physics & Philosophy: Possibly dumb but likely easy question about natural transformations inducing a functor