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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 8th 2009
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   Author: Urs
   Format: MarkdownI edited [[Trimble n-category]]: * added table of contents * added hyperlinks * moved the query boxes that seemed to contain closed discussion to the bottom. I kept the query box where I ask for a section about category theory for Trimble n-categories, but maybe we want to remove that, too. Todd has more on this on his personal web.

   I edited Trimble n-category:

   • added table of contents

   • added hyperlinks

   • moved the query boxes that seemed to contain closed discussion to the bottom. I kept the query box where I ask for a section about category theory for Trimble n-categories, but maybe we want to remove that, too. Todd has more on this on his personal web.

2. • CommentRowNumber2.
   • CommentAuthornLab edit announcer
   • CommentTimeNov 3rd 2019
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   Author: nLab edit announcer
   Format: MarkdownItexAdded journal reference to Cheng-Leinster reference. Rongmin Lu <a href="https://ncatlab.org/nlab/revision/diff/Trimble+n-category/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/Trimble+n-category/29">v29</a>, <a href="https://ncatlab.org/nlab/show/Trimble+n-category">current</a>

   Added journal reference to Cheng-Leinster reference.

   Rongmin Lu

   diff, v29, current

3. • CommentRowNumber3.
   • CommentAuthorpatrick_nicodemus1
   • CommentTimeJan 27th 2024
   • (edited Jan 27th 2024)
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   Author: patrick_nicodemus1
   Format: TextCan somebody add to this article how we get the notion of morphisms between n-categories from this? One cannot define an n+1 category without understanding what is a morphism of n-categories, we need to fully define the symmetric monoidal category of n-categories before we can explain how P_n+1 acts on it. I think that the n-lab is a bit confusing here, as this related page https://ncatlab.org/nlab/show/Batanin+omega-category says: > The definition is similar to that of Trimble n-category (which is actually a special case of a Batanin ω-category) and similar to the definition of Grothendieck-Maltsiniotis infinity-category ... When a weak ∞-category is modeled as a module over an O-operad, morphisms of modules F:C→D will correspond to strict ∞ functors. To get weak ∞-functors one has to resolve C. One way to do this is described in (Garner). It would be easy to take away from this that, since Trimble n-categories are a special case of Batanin ω-categories, the naive definition of a morphism of Trimble n-categories that we write down will be a strict morphism of n-categories. However this would be problematic as the inductive definition of Trimble (n+1)-categories should be based on a category of n-categories and weak n-morphisms, so there has to be more to the story than this. The difference between weak and strong here first appears at n=2, I think, because a tricategory has a composition pseudofunctor Hom(A,B) x Hom(B, C) -> Hom(A, C). Edit: Never mind, sorry for the confusion. I skimmed Eugenia Cheng's paper "Comparing operadic theories of n-category" and I see that in fact the definition does rely on strict morphisms between n-categories. So, a tricategory is not a Trimble 3-category, although a bicategory is a Trimble 2-category, I basically assumed that the definition was intended to subsume bicategories, tricategories and tetracategories. It would be good to add to the page here https://ncatlab.org/nlab/show/n-category whether they subsume the known notions of bicategory and tricategory.

   Can somebody add to this article how we get the notion of morphisms between n-categories from this?
   One cannot define an n+1 category without understanding what is a morphism of n-categories, we need to fully define the symmetric monoidal category of n-categories before we can explain how P_n+1 acts on it.
   
   I think that the n-lab is a bit confusing here, as this related page https://ncatlab.org/nlab/show/Batanin+omega-category says:
   > The definition is similar to that of Trimble n-category (which is actually a special case of a Batanin ω-category) and similar to the definition of Grothendieck-Maltsiniotis infinity-category ... When a weak ∞-category is modeled as a module over an O-operad, morphisms of modules F:C→D will correspond to strict ∞ functors. To get weak ∞-functors one has to resolve C. One way to do this is described in (Garner).
   
   It would be easy to take away from this that, since Trimble n-categories are a special case of Batanin ω-categories, the naive definition of a morphism of Trimble n-categories that we write down will be a strict morphism of n-categories. However this would be problematic as the inductive definition of Trimble (n+1)-categories should be based on a category of n-categories and weak n-morphisms, so there has to be more to the story than this. The difference between weak and strong here first appears at n=2, I think, because a tricategory has a composition pseudofunctor Hom(A,B) x Hom(B, C) -> Hom(A, C).
   
   Edit: Never mind, sorry for the confusion. I skimmed Eugenia Cheng's paper "Comparing operadic theories of n-category" and I see that in fact the definition does rely on strict morphisms between n-categories. So, a tricategory is not a Trimble 3-category, although a bicategory is a Trimble 2-category, I basically assumed that the definition was intended to subsume bicategories, tricategories and tetracategories. It would be good to add to the page here https://ncatlab.org/nlab/show/n-category whether they subsume the known notions of bicategory and tricategory.
4. • CommentRowNumber4.
   • CommentAuthorpatrick_nicodemus1
   • CommentTimeJan 27th 2024
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   Author: patrick_nicodemus1
   Format: TextThis is potentially pedantic but I also think that the opening line "Todd Trimble‘s definition of weak n-category is an example of a notion of weak omega-category" is confusing. Informally, a weak n-category can be regarded as a weak infinity category that is truncated above degree n. Formally, we should not call a definition of n-category a model "a notion of weak omega-category" unless it is really, transparently, a definition of an omega-category, together with a clear definition of what it means for that model of omega-category to be n-truncated, such that it is obvious that the proposed definition of n-category is an n-truncated omega category. Since in this case it is less than obvious how to extend the definition to the omega case, and since this extension does not appear to actually subsume the case of n-category, I don't think this is appropriate.

   This is potentially pedantic but I also think that the opening line "Todd Trimble‘s definition of weak n-category is an example of a notion of weak omega-category" is confusing. Informally, a weak n-category can be regarded as a weak infinity category that is truncated above degree n. Formally, we should not call a definition of n-category a model "a notion of weak omega-category" unless it is really, transparently, a definition of an omega-category, together with a clear definition of what it means for that model of omega-category to be n-truncated, such that it is obvious that the proposed definition of n-category is an n-truncated omega category. Since in this case it is less than obvious how to extend the definition to the omega case, and since this extension does not appear to actually subsume the case of n-category, I don't think this is appropriate.
5. • CommentRowNumber5.
   • CommentAuthornLab edit announcer
   • CommentTimeJan 27th 2024
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   Author: nLab edit announcer
   Format: MarkdownItexAdded a note expressing my understanding of how Trimble $n$-categories relate to bicategories and tricategories. Patrick Nicodemus <a href="https://ncatlab.org/nlab/revision/diff/Trimble+n-category/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/Trimble+n-category/31">v31</a>, <a href="https://ncatlab.org/nlab/show/Trimble+n-category">current</a>

   Added a note expressing my understanding of how Trimble $n$-categories relate to bicategories and tricategories.

   Patrick Nicodemus

   diff, v31, current

6. • CommentRowNumber6.
   • CommentAuthornLab edit announcer
   • CommentTimeJan 27th 2024
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexremoved query boxes of discussions: +--{.query} [[John Baez|John]] has just substituted "topological" for my ([[Todd Trimble|Todd]]'s) original "tautological", which I don't mind at all -- "topological" is correct and arguably more informative -- but maybe this is a good place to explain that "tautological" was not a typo, but rather an expansion of a technical usage: One of the first examples of an operad that people are taught is the one where the n-th component consists of _all_ n-ary operations $X^n \to X$. (Here $X^n$ denotes an n-fold tensor power in a monoidal category.) Because it's the simplest or most obvious example of an operad, and because it's canonically there no matter what $X$ is, it's often in the literature called a "tautological operad". What is less often noted is that by the same logic, $hom(X, X^n)$ is an equally tautological operad, except that it consists of all "co-operations" instead of operations. Then, I am applying this observation to the example of the monoidal category of topological cospans from a point to itself, where $X$ is the interval. _Toby_: Considering that the original was 'tautological topological', which is more informative, I\'ve changed it back. But perhaps this explanation has a place in the main text? _Todd_: Sounds like a good idea. If you can figure out a smooth way to do that, I'd be grateful. _Toby_: Actually, I just realised that most of it is already in the next paragraph below. I\'ll just add a sentence noting that this explains the name. =-- +--{.query} [[Urs Schreiber|Urs]]: Thanks, Todd, this is great. If I may, let me try, also to check if I am following, to see how much of this we can do while abstracting away from Top and the concrete interval object in there. It seems that we need: * a closed monoidal homotopical category $V$ (not your $V$ above, but I'll call it $V$ nevertheless); * an object $I \in V$ fitting in an internal co-graph $pt \stackrel{\sigma, \tau}{\to} I$ such that * all the end-to-end pushouts $I^{\vee n}$ exist in $V$; * all the $V$-internal hom-objects $[I, I^{\vee n}]$ are weakly equivalent to the point (the terminal object) $[I, I^{\vee n}] \stackrel{\sim}{\to} pt$; * equipped with the structure of a co-operad internal to $V$ on $\{[I,I^{\vee n}]\}$, $n \in \mathbb{N}$. Then * for $C$ any $V$-enriched homotopical category (I want to use now that $C$ is powered over $V$) we get &#8211; now let me see... &#8211; a functor $$\Pi_{\omega}: C \to Trimble-\omega-Cat.$$ Hm, let me just let it stand this way for you to either erase all of this or maybe comment on it for the moment... [[Todd Trimble|Todd]]: Yes, that's more or less right. But there is a god-given nonpermutative operad (not co-operad) structure on the graded $V$-object $\{[I, I^{\vee n}]\}$, and that's what I'm using here. If you know how the "endomorphism operad" works (as in the entry [[operad]]), then the operad here works the same way, mutatis mutandis. As for the last part: in the original definition, I didn't know how to pass to the full-fledged $\omega$-categorical definition; I just had $n$-categories, one for each $n$. But Tom and Eugenia figured out how to make "Trimble-like" $\omega$-categories work by using coalgebraic methods. I haven't thought about whether that allows a $\Pi_\omega$... But aside from these technicalities: yes, you seem to understand what this is all about. There were some comments to this effect after I gave that lecture; someone asked what you really need to do this definition very generally. Martin Hyland cried, "Not much!" _[[Urs Schreiber]]:_ Okay, thanks. I tried now to formalize this at [[interval object]]. =-- John Wick <a href="https://ncatlab.org/nlab/revision/diff/Trimble+n-category/32">diff</a>, <a href="https://ncatlab.org/nlab/revision/Trimble+n-category/32">v32</a>, <a href="https://ncatlab.org/nlab/show/Trimble+n-category">current</a>

   removed query boxes of discussions:

   +–{.query} John has just substituted “topological” for my (Todd’s) original “tautological”, which I don’t mind at all – “topological” is correct and arguably more informative – but maybe this is a good place to explain that “tautological” was not a typo, but rather an expansion of a technical usage:

   One of the first examples of an operad that people are taught is the one where the n-th component consists of all n-ary operations $X^n \to X$. (Here $X^n$ denotes an n-fold tensor power in a monoidal category.) Because it’s the simplest or most obvious example of an operad, and because it’s canonically there no matter what $X$ is, it’s often in the literature called a “tautological operad”. What is less often noted is that by the same logic, $hom(X, X^n)$ is an equally tautological operad, except that it consists of all “co-operations” instead of operations. Then, I am applying this observation to the example of the monoidal category of topological cospans from a point to itself, where $X$ is the interval.

   Toby: Considering that the original was ’tautological topological’, which is more informative, I've changed it back. But perhaps this explanation has a place in the main text?

   Todd: Sounds like a good idea. If you can figure out a smooth way to do that, I’d be grateful.

   Toby: Actually, I just realised that most of it is already in the next paragraph below. I'll just add a sentence noting that this explains the name. =–

   +–{.query} Urs: Thanks, Todd, this is great. If I may, let me try, also to check if I am following, to see how much of this we can do while abstracting away from Top and the concrete interval object in there.

   It seems that we need:

   • a closed monoidal homotopical category $V$ (not your $V$ above, but I’ll call it $V$ nevertheless);

   • an object $I \in V$ fitting in an internal co-graph $pt \stackrel{\sigma, \tau}{\to} I$ such that

     • all the end-to-end pushouts $I^{\vee n}$ exist in $V$;

     • all the $V$-internal hom-objects $[I, I^{\vee n}]$ are weakly equivalent to the point (the terminal object) $[I, I^{\vee n}] \stackrel{\sim}{\to} pt$;

     • equipped with the structure of a co-operad internal to $V$ on $\{[I,I^{\vee n}]\}$, $n \in \mathbb{N}$.

   Then

   • for $C$ any $V$-enriched homotopical category (I want to use now that $C$ is powered over $V$) we get – now let me see… – a functor $$\Pi_{\omega}: C \to Trimble-\omega-Cat.$$

   Hm, let me just let it stand this way for you to either erase all of this or maybe comment on it for the moment…

   Todd: Yes, that’s more or less right. But there is a god-given nonpermutative operad (not co-operad) structure on the graded $V$-object $\{[I, I^{\vee n}]\}$, and that’s what I’m using here. If you know how the “endomorphism operad” works (as in the entry operad), then the operad here works the same way, mutatis mutandis.

   As for the last part: in the original definition, I didn’t know how to pass to the full-fledged $\omega$-categorical definition; I just had $n$-categories, one for each $n$. But Tom and Eugenia figured out how to make “Trimble-like” $\omega$-categories work by using coalgebraic methods. I haven’t thought about whether that allows a $\Pi_\omega$… But aside from these technicalities: yes, you seem to understand what this is all about.

   There were some comments to this effect after I gave that lecture; someone asked what you really need to do this definition very generally. Martin Hyland cried, “Not much!”

   Urs Schreiber: Okay, thanks. I tried now to formalize this at interval object.

   =–

   John Wick

   diff, v32, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJan 28th 2024
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   Author: Urs
   Format: MarkdownItexThanks for looking into cleaning up this old entry, there is much room left to do so. (This entry dates from the early days of the nLab in 2009 and had been hardly touched since.)

   Thanks for looking into cleaning up this old entry, there is much room left to do so. (This entry dates from the early days of the nLab in 2009 and had been hardly touched since.)

8. • CommentRowNumber8.
   • CommentAuthorWilf Offord
   • CommentTimeJul 6th 2024
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   Author: Wilf Offord
   Format: MarkdownItexIt was stated that not every tricategory is equivalent to one with strict interchange - this is not known, to my knowledge, and Simpson's conjecture states the opposite. <a href="https://ncatlab.org/nlab/revision/diff/Trimble+n-category/33">diff</a>, <a href="https://ncatlab.org/nlab/revision/Trimble+n-category/33">v33</a>, <a href="https://ncatlab.org/nlab/show/Trimble+n-category">current</a>

   It was stated that not every tricategory is equivalent to one with strict interchange - this is not known, to my knowledge, and Simpson’s conjecture states the opposite.

   diff, v33, current