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Atrium > Mathematics, Physics & Philosophy: Having an oo-left adjoint

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- CommentRowNumber1.
- CommentAuthorAli Caglayan
- CommentTimeMar 18th 2019
- (edited Mar 19th 2019)
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Author: Ali Caglayan
Format: MarkdownItexIn category theory, a functor $U: C \to D$ can have the property of having a left adjoint iff $\forall X \in D$, $\exists FX \in C$ and $\exists \eta_X : X \to U(F X)$ such that, $\forall A \in C$ and forall $f : X \to U(A)$, $\exists ! g : F X \to A$ such that the triangle commutes: $f = U(g) \circ \eta_X$. This definition is nice because it is very minimalistic, and it can be canonically extended to construct a left adjoint functor when it holds. I suppose another way to say this is that these are the necessary and sufficient conditions for a left adjoint defined any other way. My question is: Do we have a similar minimal sort of definition for having an oo-left adjoint? I don't have any particular model of oo-cat in mind.

In category theory, a functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">U: C \to D</annotation></semantics></math>$ can have the property of having a left adjoint iff $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>X</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">\forall X \in D</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∃</mo><mi>FX</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\exists FX \in C</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∃</mo><msub><mi>η</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>F</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exists \eta_X : X \to U(F X)</annotation></semantics></math>$ such that, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>A</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\forall A \in C</annotation></semantics></math>$ and forall $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : X \to U(A)</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∃</mo><mo>!</mo><mi>g</mi><mo>:</mo><mi>F</mi><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\exists ! g : F X \to A</annotation></semantics></math>$ such that the triangle commutes: $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">f = U(g) \circ \eta_X</annotation></semantics></math>$ .

This definition is nice because it is very minimalistic, and it can be canonically extended to construct a left adjoint functor when it holds. I suppose another way to say this is that these are the necessary and sufficient conditions for a left adjoint defined any other way.

My question is: Do we have a similar minimal sort of definition for having an oo-left adjoint?

I don’t have any particular model of oo-cat in mind.
- CommentRowNumber2.
- CommentAuthorMike Shulman
- CommentTimeMar 18th 2019
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Author: Mike Shulman
Format: MarkdownItexThe answer is yes. But in order to make it precise one probably has to choose a particular model.

The answer is yes. But in order to make it precise one probably has to choose a particular model.
- CommentRowNumber3.
- CommentAuthorDmitri Pavlov
- CommentTimeMar 18th 2019
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Author: Dmitri Pavlov
Format: MarkdownItexThe first functor should probably be called U, not F. Also A∈C, not A∈X. The condition after "such that" can be reformulated by saying that the canonical functor FX/C→X/U induced by U and precomposition with η_X is an equivalence of categories. Presumably, replacing equivalences with ∞-equivalences should yield the desired outcome.

The first functor should probably be called U, not F. Also A∈C, not A∈X.

The condition after “such that” can be reformulated by saying that the canonical functor FX/C→X/U induced by U and precomposition with η_X is an equivalence of categories.

Presumably, replacing equivalences with ∞-equivalences should yield the desired outcome.
- CommentRowNumber4.
- CommentAuthorAli Caglayan
- CommentTimeMay 26th 2019
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Author: Ali Caglayan
Format: MarkdownItexSo I had a chat with Emily and it turns out there is a way to make this precise: A functor $U : C \to D$ has a left adjoint if for all $X$ in $D$, the comma category $X / U$ has an initial object $(FX, \eta_X)$. Then you should be able to stick an oo infront of everything. This is proposition 4.1.5 in [her new book](http://www.math.jhu.edu/~eriehl/elements.pdf).

So I had a chat with Emily and it turns out there is a way to make this precise:

A functor $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">U : C \to D</annotation></semantics></math>$ has a left adjoint if for all $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>$ , the comma category $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">X / U</annotation></semantics></math>$ has an initial object $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>FX</mi><mo>,</mo><msub><mi>η</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(FX, \eta_X)</annotation></semantics></math>$ .

Then you should be able to stick an oo infront of everything.

This is proposition 4.1.5 in her new book.
- CommentRowNumber5.
- CommentAuthorAli Caglayan
- CommentTimeMay 26th 2019
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Author: Ali Caglayan
Format: MarkdownItexAm I correct in thinking that because we are considering oo-initial objects, we typically don't even need to consider higher coherences?

Am I correct in thinking that because we are considering oo-initial objects, we typically don’t even need to consider higher coherences?
- CommentRowNumber6.
- CommentAuthorMike Shulman
- CommentTimeMay 28th 2019
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Author: Mike Shulman
Format: MarkdownItexThat probably depends on your definition of $\infty$-initial...

That probably depends on your definition of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>$ -initial…

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Atrium > Mathematics, Physics & Philosophy: Having an oo-left adjoint