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Atrium > Mathematics, Physics & Philosophy: Higher arities, parametric left adjoints, and diagonal functors

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- CommentRowNumber1.
- CommentAuthorHarry Gindi
- CommentTimeSep 3rd 2011
- (edited Sep 3rd 2011)
- PermaLink
Author: Harry Gindi
Format: MarkdownItexLet $C$ be a presheaf category, and let $F : C^{m+1+n} \to C$ be a functor such that for any family of $m+n$ objects of $C$, $A_1,\dots,A_{m-i}$ and $B_1,\dots,B_{n+i}$ (letting $i$ vary between $0$ and $m$), $F(A_1,\dots,A_{m-i},-,B_1,\dots,B_{n+i})$ is a parametric left adjoint, that is, the induced functor $$L:C\to F(A_1,\dots,A_{m-i},\emptyset,B_1,\dots,B_{n+i})\downarrow C$$ admits a right adjoint. Let $\Delta^{m+1+n}:C\to C^{m+1+n}$ denote the $m+1+n$-fold diagonal functor. Let $G=F\circ \Delta^{m+1+n}$. Then is it the case that the induced functor $L_\Delta:C\to G(\emptyset)\downarrow C$ admits a right adjoint?

Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ be a presheaf category, and let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>n</mi></mrow></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : C^{m+1+n} \to C</annotation></semantics></math>$ be a functor such that for any family of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m+n</annotation></semantics></math>$ objects of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_1,\dots,A_{m-i}</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>B</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">B_1,\dots,B_{n+i}</annotation></semantics></math>$ (letting $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>$ vary between $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math>$ ), $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>i</mi></mrow></msub><mo>,</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo>,</mo><msub><mi>B</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(A_1,\dots,A_{m-i},-,B_1,\dots,B_{n+i})</annotation></semantics></math>$ is a parametric left adjoint, that is, the induced functor
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>i</mi></mrow></msub><mo>,</mo><mi>∅</mi><mo>,</mo><msub><mi>B</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">↓</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">L:C\to F(A_1,\dots,A_{m-i},\emptyset,B_1,\dots,B_{n+i})\downarrow C</annotation></semantics></math>$
admits a right adjoint.

Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Δ</mi> <mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>n</mi></mrow></msup><mo>:</mo><mi>C</mi><mo>→</mo><msup><mi>C</mi> <mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\Delta^{m+1+n}:C\to C^{m+1+n}</annotation></semantics></math>$ denote the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m+1+n</annotation></semantics></math>$ -fold diagonal functor. Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mi>F</mi><mo>∘</mo><msup><mi>Δ</mi> <mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">G=F\circ \Delta^{m+1+n}</annotation></semantics></math>$ . Then is it the case that the induced functor

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mi>Δ</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><mi>∅</mi><mo stretchy="false">)</mo><mo stretchy="false">↓</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">L_\Delta:C\to G(\emptyset)\downarrow C</annotation></semantics></math>$

admits a right adjoint?

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Atrium > Mathematics, Physics & Philosophy: Higher arities, parametric left adjoints, and diagonal functors