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1. • CommentRowNumber1.
   • CommentAuthorJohn Baez
   • CommentTimeJan 22nd 2018
   • (edited Jan 22nd 2018)
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   Author: John Baez
   Format: MarkdownItexThis is probably a request for Todd! Over on [[colimits for categories of algebras]] there's a corollary I really need right now, about Eilerberg-Moore categories being cocomplete, and the remark: > The hypotheses of the preceding corollary hold when $C$ is a complete, cocomplete, cartesian closed category and $C$ is the monad corresponding to a [[finitary algebraic theory]]. That sounds like exactly what I want, but when I click on [finitary algebraic theory](https://ncatlab.org/nlab/show/algebraic+theory) I get taken to a page that doesn't have the definition of "finitary algebraic theory". I think I know what this means, so I could guess and stick it in, but I think I should let the expert do it. Oh, whoops! - as usual, I actually need a multi-sorted generalization. But still it would be nice to have this clarified.

   This is probably a request for Todd!

   Over on colimits for categories of algebras there’s a corollary I really need right now, about Eilerberg-Moore categories being cocomplete, and the remark:

   The hypotheses of the preceding corollary hold when $C$ is a complete, cocomplete, cartesian closed category and $C$ is the monad corresponding to a finitary algebraic theory.

   That sounds like exactly what I want, but when I click on finitary algebraic theory I get taken to a page that doesn’t have the definition of “finitary algebraic theory”. I think I know what this means, so I could guess and stick it in, but I think I should let the expert do it.

   Oh, whoops! - as usual, I actually need a multi-sorted generalization. But still it would be nice to have this clarified.

2. • CommentRowNumber2.
   • CommentAuthorJohn Baez
   • CommentTimeJan 23rd 2018
   • (edited Jan 23rd 2018)
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   Author: John Baez
   Format: MarkdownItexOh, goodie - the result I need is exactly Prop. 2.1 on Todd's page [multisorted Lawvere theories](https://ncatlab.org/toddtrimble/published/multisorted+Lawvere+theories)! Yet again you saved me, Todd! (But still it'd be good to define "finitary algebraic theory". I find this slightly nerve-racking because my usual "algebraic theories" are, I believe, just your finitary ones.)

   Oh, goodie - the result I need is exactly Prop. 2.1 on Todd’s page multisorted Lawvere theories! Yet again you saved me, Todd!

   (But still it’d be good to define “finitary algebraic theory”. I find this slightly nerve-racking because my usual “algebraic theories” are, I believe, just your finitary ones.)

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 23rd 2018
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   Author: Todd_Trimble
   Format: MarkdownItexGlad you figured it out on your own, John. :-) It seems like "finitary algebraic theory" ought to redirect to [[Lawvere theory]]. Let me look into it. The bulk of [[algebraic theory]] is really mostly about the various infinitary versions.

   Glad you figured it out on your own, John. :-) It seems like “finitary algebraic theory” ought to redirect to Lawvere theory. Let me look into it. The bulk of algebraic theory is really mostly about the various infinitary versions.

4. • CommentRowNumber4.
   • CommentAuthorJohn Baez
   • CommentTimeJan 23rd 2018
   • (edited Jan 23rd 2018)
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   Author: John Baez
   Format: MarkdownItexAnother thing I find mildly nerve-racking is "finitary algebraic theory" versus "Lawvere theory" and "multi-sorted Lawvere theory". We'll never get everyone in the universe to agree on how these terms are used, but a surprisingly large number of grad students take the nLab as gospel, so if the nLab decides on some clear and consistent usages, most mathematicians (at least on Earth) are likely to fall in line over time.

   Another thing I find mildly nerve-racking is “finitary algebraic theory” versus “Lawvere theory” and “multi-sorted Lawvere theory”. We’ll never get everyone in the universe to agree on how these terms are used, but a surprisingly large number of grad students take the nLab as gospel, so if the nLab decides on some clear and consistent usages, most mathematicians (at least on Earth) are likely to fall in line over time.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJan 23rd 2018
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   Author: Mike Shulman
   Format: MarkdownItexRedirecting [[finitary algebraic theory]] to [[Lawvere theory]] seems like a good start. What's the remaining confusion? Is it whether a "finitary algebraic theory" is necessarily single-sorted? Or whether it should refer to the syntactic presentation instead of the semantic incarnation?

   Redirecting finitary algebraic theory to Lawvere theory seems like a good start. What’s the remaining confusion? Is it whether a “finitary algebraic theory” is necessarily single-sorted? Or whether it should refer to the syntactic presentation instead of the semantic incarnation?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJan 23rd 2018
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   Author: Mike Shulman
   Format: MarkdownItexOr whether an "algebraic theory" should be finitary by default? I expect that's the traditional usage among non-category-theoretic universal-algebraists. On the other hand I believe there's a certain tradition that refers to any category with a monadic forgetful functor as "algebraic". Then again, an "essentially algebraic theory" seems fairly unambiguously finitary, so if an "algebraic theory" is not necessarily finitary then we wouldn't have algebraic $\Rightarrow$ essentially algebraic, which is arguably a [[red herring principle|red herring]].

   Or whether an “algebraic theory” should be finitary by default? I expect that’s the traditional usage among non-category-theoretic universal-algebraists. On the other hand I believe there’s a certain tradition that refers to any category with a monadic forgetful functor as “algebraic”. Then again, an “essentially algebraic theory” seems fairly unambiguously finitary, so if an “algebraic theory” is not necessarily finitary then we wouldn’t have algebraic $\Rightarrow$ essentially algebraic, which is arguably a red herring.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 23rd 2018
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   Author: Todd_Trimble
   Format: MarkdownItexGood questions, all. I think getting some terminological clarity here is a task that I can ( at least help) undertake, starting this evening my time. There's another point of view that would take an algebraic theory (putting aside finitary or not) to mean a certain presentation in terms of operations and axioms. (What we call a Lawvere theory or a clone is kind of a maximally saturated such presentation which takes on all definable operations at once.) Category theorists and we nLab denizens often de-emphasize particular presentations, but it's certainly alive and well in model-theoretic contexts and we should be sensitive to that also.

   Good questions, all. I think getting some terminological clarity here is a task that I can ( at least help) undertake, starting this evening my time.

   There’s another point of view that would take an algebraic theory (putting aside finitary or not) to mean a certain presentation in terms of operations and axioms. (What we call a Lawvere theory or a clone is kind of a maximally saturated such presentation which takes on all definable operations at once.) Category theorists and we nLab denizens often de-emphasize particular presentations, but it’s certainly alive and well in model-theoretic contexts and we should be sensitive to that also.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJan 23rd 2018
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   Author: Mike Shulman
   Format: MarkdownItexThanks Todd! Yes, that's what I meant to refer to by a "syntactic presentation". I'm personally [very sympathetic](https://golem.ph.utexas.edu/category/2010/07/what_is_a_theory.html) to that view.

   Thanks Todd! Yes, that’s what I meant to refer to by a “syntactic presentation”. I’m personally very sympathetic to that view.