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nLab > Latest Changes: lax-idempotent 2-monad

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- CommentRowNumber1.
- CommentAuthorJohn Baez
- CommentTimeAug 14th 2020
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Author: John Baez
Format: MarkdownItexI corrected an apparent typo: A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta \circ a$ to A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta_A \circ a$ It might be nice to say $\eta_A$ is the unit of the algebra.... <a href="https://ncatlab.org/nlab/revision/diff/lax-idempotent+2-monad/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/lax-idempotent+2-monad/22">v22</a>, <a href="https://ncatlab.org/nlab/show/lax-idempotent+2-monad">current</a>

I corrected an apparent typo:

A 2-monad $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>$ as above is lax-idempotent if and only if for any $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>$ -algebra $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo lspace="0.11111em">:</mo><mi>T</mi><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \colon T A \to A</annotation></semantics></math>$ there is a 2-cell $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>θ</mi> <mi>a</mi></msub><mo lspace="0.11111em">:</mo><mn>1</mn><mo>⇒</mo><mi>η</mi><mo>∘</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\theta_a \colon 1 \Rightarrow \eta \circ a</annotation></semantics></math>$

to

A 2-monad $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>$ as above is lax-idempotent if and only if for any $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>$ -algebra $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo lspace="0.11111em">:</mo><mi>T</mi><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \colon T A \to A</annotation></semantics></math>$ there is a 2-cell $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>θ</mi> <mi>a</mi></msub><mo lspace="0.11111em">:</mo><mn>1</mn><mo>⇒</mo><msub><mi>η</mi> <mi>A</mi></msub><mo>∘</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\theta_a \colon 1 \Rightarrow \eta_A \circ a</annotation></semantics></math>$

It might be nice to say $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\eta_A</annotation></semantics></math>$ is the unit of the algebra….

diff, v22, current
- CommentRowNumber2.
- CommentAuthorzskoda
- CommentTimeSep 27th 2022
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Author: zskoda
Format: MarkdownItexFor the reference * [[Anders Kock]], _Monads for which structures are adjoint to units_ , Aarhus Preprint 1972/73 No. 35 <http://home.imf.au.dk/kock/msau1.PDF> it seems that the pdf link does not work any more. It is also referred as such at author's webpage.
For the reference
- Anders Kock, Monads for which structures are adjoint to units , Aarhus Preprint 1972/73 No. 35 http://home.imf.au.dk/kock/msau1.PDF
it seems that the pdf link does not work any more. It is also referred as such at author’s webpage.
- CommentRowNumber3.
- CommentAuthoranuyts
- CommentTimeFeb 28th 2023
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Author: anuyts
Format: MarkdownItexLink to lax-idempotent 2-adjunctions for equivalent conditions. <a href="https://ncatlab.org/nlab/revision/diff/lax-idempotent+2-monad/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/lax-idempotent+2-monad/27">v27</a>, <a href="https://ncatlab.org/nlab/show/lax-idempotent+2-monad">current</a>

Link to lax-idempotent 2-adjunctions for equivalent conditions.

diff, v27, current
- CommentRowNumber4.
- CommentAuthorvarkor
- CommentTimeApr 22nd 2023
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Author: varkor
Format: MarkdownItexAdd redirects for the "pseudomonad" notions. <a href="https://ncatlab.org/nlab/revision/diff/lax-idempotent+2-monad/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/lax-idempotent+2-monad/29">v29</a>, <a href="https://ncatlab.org/nlab/show/lax-idempotent+2-monad">current</a>

Add redirects for the “pseudomonad” notions.

diff, v29, current
- CommentRowNumber5.
- CommentAuthorvarkor
- CommentTimeJun 23rd 2023
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Author: varkor
Format: MarkdownItexMention the adjointness characterisation. <a href="https://ncatlab.org/nlab/revision/diff/lax-idempotent+2-monad/30">diff</a>, <a href="https://ncatlab.org/nlab/revision/lax-idempotent+2-monad/30">v30</a>, <a href="https://ncatlab.org/nlab/show/lax-idempotent+2-monad">current</a>

Mention the adjointness characterisation.

diff, v30, current
- CommentRowNumber6.
- CommentAuthorvarkor
- CommentTimeOct 17th 2024
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Author: varkor
Format: MarkdownItexAdded another reference of Marmolejo. <a href="https://ncatlab.org/nlab/revision/diff/lax-idempotent+2-monad/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/lax-idempotent+2-monad/35">v35</a>, <a href="https://ncatlab.org/nlab/show/lax-idempotent+2-monad">current</a>

Added another reference of Marmolejo.

diff, v35, current
- CommentRowNumber7.
- CommentAuthornLab edit announcer
- CommentTimeMar 20th 2025
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Author: nLab edit announcer
Format: MarkdownItexI noticed while working out the definition that the condition was ported from the lax idempotent adjunction page, but the case for monads is simpler since we already have that $\mu_A \circ T \eta_A$ is iso to the identity. Quentin <a href="https://ncatlab.org/nlab/revision/diff/lax-idempotent+2-monad/36">diff</a>, <a href="https://ncatlab.org/nlab/revision/lax-idempotent+2-monad/36">v36</a>, <a href="https://ncatlab.org/nlab/show/lax-idempotent+2-monad">current</a>

I noticed while working out the definition that the condition was ported from the lax idempotent adjunction page, but the case for monads is simpler since we already have that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi> <mi>A</mi></msub><mo>∘</mo><mi>T</mi><msub><mi>η</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\mu_A \circ T \eta_A</annotation></semantics></math>$ is iso to the identity.

Quentin

diff, v36, current
- CommentRowNumber8.
- CommentAuthorvarkor
- CommentTimeJul 9th 2025
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Author: varkor
Format: MarkdownItexAdded a definition of lax-idempotent 2-comonad. <a href="https://ncatlab.org/nlab/revision/diff/lax-idempotent+2-monad/38">diff</a>, <a href="https://ncatlab.org/nlab/revision/lax-idempotent+2-monad/38">v38</a>, <a href="https://ncatlab.org/nlab/show/lax-idempotent+2-monad">current</a>

Added a definition of lax-idempotent 2-comonad.

diff, v38, current

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nLab > Latest Changes: lax-idempotent 2-monad