Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: faithful functor

Bottom of Page

1 to 6 of 6

- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeSep 9th 2018
- PermaLink
Author: Urs
Format: MarkdownItexthere was a pointer to the generalization of "faithful functor" to 2-categories. I have added below that pointer to the corresponding version for $(\infty,1)$-categories. <a href="https://ncatlab.org/nlab/revision/diff/faithful+functor/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/faithful+functor/11">v11</a>, <a href="https://ncatlab.org/nlab/show/faithful+functor">current</a>

there was a pointer to the generalization of “faithful functor” to 2-categories. I have added below that pointer to the corresponding version for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>$ -categories.

diff, v11, current
- CommentRowNumber2.
- CommentAuthorGrant_Bradley
- CommentTimeJul 2nd 2021
- PermaLink
Author: Grant_Bradley
Format: MarkdownItexadded properties of faithful functor in terms of hom-sets <a href="https://ncatlab.org/nlab/revision/diff/faithful+functor/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/faithful+functor/13">v13</a>, <a href="https://ncatlab.org/nlab/show/faithful+functor">current</a>

added properties of faithful functor in terms of hom-sets

diff, v13, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeJul 2nd 2021
- PermaLink
Author: Urs
Format: MarkdownItexIf it wasn't clear the way it was, then I suggest to expand on it while sticking to the notation that the paragraph starts with. I have made it show this ([here](https://ncatlab.org/nlab/show/faithful+functor#In1CategoryTheory)): $$ \array{ \mathcal{C}(x,y) &\xhookrightarrow{\;\; F_{x,y} \;\;}& \mathcal{D}(F(x), F(y)) \\ (x \overset{\phi}{\to} y) &\mapsto& \big( F(x)\overset{F(\phi)}{\to} F(y) \big) \,. } $$ Also I have removed the side-remark you added that faithful functors are called "embeddings", because that terminology is fraught with issues. Instead I have added under "Related concepts" a pointer to *[[embedding of categories]]*, where this is discussed in more detail. <a href="https://ncatlab.org/nlab/revision/diff/faithful+functor/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/faithful+functor/14">v14</a>, <a href="https://ncatlab.org/nlab/show/faithful+functor">current</a>

If it wasn’t clear the way it was, then I suggest to expand on it while sticking to the notation that the paragraph starts with. I have made it show this (here):
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↪</mo><mrow><mspace width="0.27778em"/><mspace width="0.27778em"/><msub><mi>F</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mspace width="0.27778em"/><mspace width="0.27778em"/></mrow></mover></mtd> <mtd><mi>𝒟</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mover><mo>→</mo><mi>ϕ</mi></mover><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="0.16667em"/><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}(x,y) &amp;\xhookrightarrow{\;\; F_{x,y} \;\;}&amp; \mathcal{D}(F(x), F(y)) \\ (x \overset{\phi}{\to} y) &amp;\mapsto&amp; \big( F(x)\overset{F(\phi)}{\to} F(y) \big) \,. } </annotation></semantics></math>$
Also I have removed the side-remark you added that faithful functors are called “embeddings”, because that terminology is fraught with issues. Instead I have added under “Related concepts” a pointer to embedding of categories, where this is discussed in more detail.

diff, v14, current
- CommentRowNumber4.
- CommentAuthorHurkyl
- CommentTimeJul 2nd 2021
- (edited Jul 2nd 2021)
- PermaLink
Author: Hurkyl
Format: MarkdownItexI'm not sure that 0-truncated morphisms are the right thing to list under generalizations of the notion of faithful functor. It seems a reasonable characterization for functors between infinity groupoids, since $X \to Y$ being 0-truncated means $X(x,x) = \Omega_x X \to \Omega_y Y = Y(y,y)$ is a monomorphism. But it doesn't seem reasonable for functors between infinity categories. For example, if $C$ is a 1-category with no nontrivial isomorphisms, then when viewing $C$ as an infinity category, $C \to *$ is a 0-truncated morphism of $\infty Cat$ (since the nerve of $C$ is a complete segal space whose levels are all sets). But $C \to *$ need not be a faithful functor in the sence of 1-categories. I would expect "locally monic" would be the right generalization to infinity categories, and dually "locally an effective epimorphism" would be the right generalization of "full".

I’m not sure that 0-truncated morphisms are the right thing to list under generalizations of the notion of faithful functor.

It seems a reasonable characterization for functors between infinity groupoids, since $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math>$ being 0-truncated means $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi><mo>→</mo><msub><mi>Ω</mi> <mi>y</mi></msub><mi>Y</mi><mo>=</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(x,x) = \Omega_x X \to \Omega_y Y = Y(y,y)</annotation></semantics></math>$ is a monomorphism.

But it doesn’t seem reasonable for functors between infinity categories. For example, if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ is a 1-category with no nontrivial isomorphisms, then when viewing $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ as an infinity category, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C \to *</annotation></semantics></math>$ is a 0-truncated morphism of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\infty Cat</annotation></semantics></math>$ (since the nerve of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>$ is a complete segal space whose levels are all sets). But $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C \to *</annotation></semantics></math>$ need not be a faithful functor in the sence of 1-categories.

I would expect “locally monic” would be the right generalization to infinity categories, and dually “locally an effective epimorphism” would be the right generalization of “full”.
- CommentRowNumber5.
- CommentAuthorMike Shulman
- CommentTimeJul 2nd 2021
- PermaLink
Author: Mike Shulman
Format: MarkdownItexMaybe just say that 0-truncated morphisms in an $(\infty,1)$-category are a generalization of faithful functors between groupoids?

Maybe just say that 0-truncated morphisms in an $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>$ -category are a generalization of faithful functors between groupoids?
- CommentRowNumber6.
- CommentAuthorHurkyl
- CommentTimeJul 2nd 2021
- PermaLink
Author: Hurkyl
Format: MarkdownItexI understand the motivation of the existing text, and have crafted a suitable warning. Aside: is there any reason we shouldn't define a faithful functor between (∞,1)-categories to be one that is locally monic? <a href="https://ncatlab.org/nlab/revision/diff/faithful+functor/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/faithful+functor/15">v15</a>, <a href="https://ncatlab.org/nlab/show/faithful+functor">current</a>

I understand the motivation of the existing text, and have crafted a suitable warning.

Aside: is there any reason we shouldn’t define a faithful functor between (∞,1)-categories to be one that is locally monic?

diff, v15, current

1 to 6 of 6

nForum

Discussion Feed

Not signed in

Site Tag Cloud

nLab > Latest Changes: faithful functor