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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 16th 2015
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   Author: Todd_Trimble
   Format: MarkdownItexIn [[model]] I find the following sentence: > For $Syn(T)$ the [[syntactic category]] of a [[Lawvere theory]], and for $C$ any category with finite [[limit]]s, a _model_ for $T$ in $C$ is a product-preserving functor $$ N : Syn(T) \to C \,. $$ To my way of thinking, the Lawvere theory $T$ really *is* the syntactic category: the bottom line is that $T$ (or the syntactic category or the category of contexts or whatever you want to call it) is characterized as the 2-initial object in the 2-category of categories with finite products that come equipped with a model of the desired type. My concern is that readers might be led to suppose that the syntactic category is a secondary construction one has to apply to a Lawvere theory, and also the pages on [[context]] and [[syntactic category]] are also in a less than optimal state and the reader could come away thinking that all this is more complicated than it really is, if you see what I mean.

   In model I find the following sentence:

   For $Syn(T)$ the syntactic category of a Lawvere theory, and for $C$ any category with finite limits, a model for $T$ in $C$ is a product-preserving functor

   $$N : Syn(T) \to C \,.$$

   To my way of thinking, the Lawvere theory $T$ really is the syntactic category: the bottom line is that $T$ (or the syntactic category or the category of contexts or whatever you want to call it) is characterized as the 2-initial object in the 2-category of categories with finite products that come equipped with a model of the desired type.

   My concern is that readers might be led to suppose that the syntactic category is a secondary construction one has to apply to a Lawvere theory, and also the pages on context and syntactic category are also in a less than optimal state and the reader could come away thinking that all this is more complicated than it really is, if you see what I mean.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 16th 2015
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   Author: Urs
   Format: MarkdownItexThe rationale was that in the general context of theories and models, we make a distinction between a theory and its syntactic category, and so for consistency of the presentation this should also be made also for Lawvere theories. Of course historically somehow "Lawvere theory" got to be the term for the syntactic category of an algebraic theory. I'll be happy with whatever you change it to.

   The rationale was that in the general context of theories and models, we make a distinction between a theory and its syntactic category, and so for consistency of the presentation this should also be made also for Lawvere theories. Of course historically somehow “Lawvere theory” got to be the term for the syntactic category of an algebraic theory. I’ll be happy with whatever you change it to.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 16th 2015
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   Author: Todd_Trimble
   Format: MarkdownItexI see; thanks. Regarding "Of course historically somehow" -- there's no mystery, of course, because as you know that is precisely one of the seminal ideas introduced in Lawvere's thesis, that a "theory" is most usefully and elegantly presented as a structured (syntactic) category. Anyhow, I take your point.

   I see; thanks. Regarding “Of course historically somehow” – there’s no mystery, of course, because as you know that is precisely one of the seminal ideas introduced in Lawvere’s thesis, that a “theory” is most usefully and elegantly presented as a structured (syntactic) category. Anyhow, I take your point.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 16th 2015
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   Author: Todd_Trimble
   Format: MarkdownItexI just changed the formulation slightly.

   I just changed the formulation slightly.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 16th 2015
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   Author: Urs
   Format: MarkdownItexWhile I am happy with this identification, the distinction between theories and their syntactic categories is something that in large corners of the community is not taken for granted. It is for instance an ongoing matter of discussion in the HoTT community. Maybe when you edit the entry (if that's what you are about to do) you could, once you have changed it to your preferred wording, make sure that there appears a remark which highlights that while theories and their syntactic categories are not identical concepts, they may be nevertheless be equivalent (when suitably set up).

   While I am happy with this identification, the distinction between theories and their syntactic categories is something that in large corners of the community is not taken for granted. It is for instance an ongoing matter of discussion in the HoTT community.

   Maybe when you edit the entry (if that’s what you are about to do) you could, once you have changed it to your preferred wording, make sure that there appears a remark which highlights that while theories and their syntactic categories are not identical concepts, they may be nevertheless be equivalent (when suitably set up).

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 16th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, our messages overlapped. Okay, thanks!

   Ah, our messages overlapped. Okay, thanks!

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 16th 2015
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   Author: Mike Shulman
   Format: MarkdownItexRelevant historical discussion: <https://golem.ph.utexas.edu/category/2010/07/what_is_a_theory.html>

   Relevant historical discussion: https://golem.ph.utexas.edu/category/2010/07/what_is_a_theory.html

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 17th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThanks for bringing up that thread, Mike. I think I only lightly skimmed it the first time around (which I am badly in the habit of doing, I'm afraid); it was enjoyable reading this time around. At some point there was a mini-revelation of how different conceptions of theory could be stratified according to the level (sorts, operations, axioms) where one considers taking closures, and there was some brief discussion of what terminology would be good ($k$-axiomatized, $k$-embodied, etc.). Did you come up with something in the end (even privately?). A possibility that occurs to me here is "$k$-saturation". For example, I think of a Lawvere theory as a kind of saturation of a traditional theory by closing up under all definable operations to obtain something unbiased. [Similarly, I think of the (usual method of) strictification of a monoidal category as a saturation under cliques or ana-objects to obtain something unbiased.] Something about "$k$-saturated" rolls off my tongue maybe a little more easily (maybe because you don't have two vowel sounds in quick succession as one does with "$k$-axiomatized").

   Thanks for bringing up that thread, Mike. I think I only lightly skimmed it the first time around (which I am badly in the habit of doing, I’m afraid); it was enjoyable reading this time around.

   At some point there was a mini-revelation of how different conceptions of theory could be stratified according to the level (sorts, operations, axioms) where one considers taking closures, and there was some brief discussion of what terminology would be good ($k$-axiomatized, $k$-embodied, etc.). Did you come up with something in the end (even privately?). A possibility that occurs to me here is “$k$-saturation”. For example, I think of a Lawvere theory as a kind of saturation of a traditional theory by closing up under all definable operations to obtain something unbiased. [Similarly, I think of the (usual method of) strictification of a monoidal category as a saturation under cliques or ana-objects to obtain something unbiased.] Something about “$k$-saturated” rolls off my tongue maybe a little more easily (maybe because you don’t have two vowel sounds in quick succession as one does with “$k$-axiomatized”).

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 17th 2015
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   Author: Mike Shulman
   Format: MarkdownItexNo, I don't remember deciding on any terminology for that.

   No, I don’t remember deciding on any terminology for that.