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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 20th 2010
   • (edited Oct 20th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexSo I finally realized something in Leinster's book. After all of the trouble we go to constructing weak n-categories, we are left with the somewhat unsatisfying result that there is (was?) no good notion of a weak n-functor for higher n, at least for existing operadic models. At the moment, (and please correct me if I'm wrong!) it seems like the Segal-style models of n-categories ($(\infty,n)$ technically) are the only ones that give a straithforward definition of a higher functor (note: The $\infty,1$ case is irrelevant. I'm specifically talking about models for weak n-categories). If I'm wrong, has anyone figured out a way to define a weak n-functor between the n-categories of Batanin or Leinster?

   So I finally realized something in Leinster’s book. After all of the trouble we go to constructing weak n-categories, we are left with the somewhat unsatisfying result that there is (was?) no good notion of a weak n-functor for higher n, at least for existing operadic models.

   At the moment, (and please correct me if I’m wrong!) it seems like the Segal-style models of n-categories ($(\infty,n)$ technically) are the only ones that give a straithforward definition of a higher functor (note: The $\infty,1$ case is irrelevant. I’m specifically talking about models for weak n-categories).

   If I’m wrong, has anyone figured out a way to define a weak n-functor between the n-categories of Batanin or Leinster?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 20th 2010
   • (edited Oct 20th 2010)
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   Author: Urs
   Format: MarkdownItexYou are right: this is a technical problem with most algebraic definitions of higher categories: in terms of model category theory imagery (which is precise in many cases and hence conveys the problem well) these tend to produce models for weak $n$-categories that are fibrant, but not cofibrant. A weak $n$-functor would be an ordinary morphism out of a cofibrant replacement, but it may be hard or unknown how to get these replacements, hence how to weaken the $n$-functors. I know one proposal on how to do it right. That's by Todd. You can see it on his personal web [[toddtrimble:Further developments on Trimble n-categories]] Hm, let me see, now that I look at this page, it does not contain some material that Todd once sent me by email. The web version seems to break off before it gets to the n-categories.

   You are right: this is a technical problem with most algebraic definitions of higher categories:

   in terms of model category theory imagery (which is precise in many cases and hence conveys the problem well) these tend to produce models for weak $n$-categories that are fibrant, but not cofibrant. A weak $n$-functor would be an ordinary morphism out of a cofibrant replacement, but it may be hard or unknown how to get these replacements, hence how to weaken the $n$-functors.

   I know one proposal on how to do it right. That’s by Todd. You can see it on his personal web Further developments on Trimble n-categories (toddtrimble)

   Hm, let me see, now that I look at this page, it does not contain some material that Todd once sent me by email. The web version seems to break off before it gets to the n-categories.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 21st 2010
   • (edited Oct 21st 2010)
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   Author: Urs
   Format: MarkdownItexI don't seem to be able to find the email that Todd once sent with the detailed account. So we have to wait for Todd for this. But the basic idea was this: Let $O$ be the operad such that $O$-modules ("$O$-algebras") are weak $n$-categories. Construct a _contractible_ $O$-module $P$ by a bar construction. This plays the role of the cofibrant replacement of the point $n$-category. Then define a weak $n$-functor $C \to D$ beween weak $n$-categories $C, D \in O Mod$ as a morphism of $O$-modules $$ C \otimes_O P \to D \,, $$ where on the left we take the tensor product of $O$-modules. So $C \otimes_O P$ here plas the role of a cofibrant replacement for $C$. I forget what the details of the status of this were. The point resolution $P$ here was a very specific one, related to Stasheff's associahedra somehow.

   I don’t seem to be able to find the email that Todd once sent with the detailed account. So we have to wait for Todd for this.

   But the basic idea was this: Let $O$ be the operad such that $O$-modules (“$O$-algebras”) are weak $n$-categories.

   Construct a contractible $O$-module $P$ by a bar construction. This plays the role of the cofibrant replacement of the point $n$-category.

   Then define a weak $n$-functor $C \to D$ beween weak $n$-categories $C, D \in O Mod$ as a morphism of $O$-modules

   $$C \otimes_O P \to D \,,$$

   where on the left we take the tensor product of $O$-modules.

   So $C \otimes_O P$ here plas the role of a cofibrant replacement for $C$.

   I forget what the details of the status of this were. The point resolution $P$ here was a very specific one, related to Stasheff’s associahedra somehow.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 21st 2010
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   Author: DavidRoberts
   Format: MarkdownItex@Urs, I presume this is related to 'your' $n$-anafunctors? As in, recovering weak 2-functors from spans of strict 2-functors... It would be interesting to check if $C\otimes_O P$ was related to your resolution of the domain 2-category.

   @Urs,

   I presume this is related to ’your’ $n$-anafunctors? As in, recovering weak 2-functors from spans of strict 2-functors… It would be interesting to check if $C\otimes_O P$ was related to your resolution of the domain 2-category.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 21st 2010
   • (edited Oct 21st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWell, you mean for strict $\omega$-categories? Yes, that's what got me started back then: looking at strict $\omega$-categories, the evident morphisms are also strict $\omega$-functors. But we have the "[[folk model structure]]" on $Str \omega Cat$ and this models strict $\omega$-categories with _weak_ $\omega$-functors between them by modelling a weak $\omega$-functor $f : C \to D$ as a strict one out of a cofibrant resolution $$ \array{ \hat C &\to& D \\ {}^{\mathllap{\simeq}}\downarrow \\ C } \,. $$ This actually goes a long way when one looks at sheaves with values in $\infty$-categories. There are many interesting strict $\omega$-groupoid valued sheaves whose geoemtric relaization is far from being a strict $\infty$-groupoid. These can be modeled conveniently with this kind of technology.

   Well, you mean for strict $\omega$-categories? Yes, that’s what got me started back then:

   looking at strict $\omega$-categories, the evident morphisms are also strict $\omega$-functors. But we have the “folk model structure” on $Str \omega Cat$ and this models strict $\omega$-categories with weak $\omega$-functors between them by modelling a weak $\omega$-functor $f : C \to D$ as a strict one out of a cofibrant resolution

   $$\array{ \hat C &\to& D \\ {}^{\mathllap{\simeq}}\downarrow \\ C } \,.$$

   This actually goes a long way when one looks at sheaves with values in $\infty$-categories. There are many interesting strict $\omega$-groupoid valued sheaves whose geoemtric relaization is far from being a strict $\infty$-groupoid. These can be modeled conveniently with this kind of technology.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeOct 21st 2010
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   Author: Mike Shulman
   Format: MarkdownItex[Here](http://www.comp.mq.edu.au/~rgarner/Womaps/Womaps.html) is another approach.

   Here is another approach.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeOct 21st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, nice. I wasn't aware of this. This should be recorded in some $n$Lab entry.

   Ah, nice. I wasn’t aware of this. This should be recorded in some $n$Lab entry.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 21st 2010
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   Author: Todd_Trimble
   Format: MarkdownItexI'm here. It seems Richard Garner got farther with this implementing this idea than I did, but Urs has the right spirit. The idea is to use a free contractible $P$-bimodule $B$, and define a weak functor $X \to Y$ to be a strict functor (a strict $P$-algebra map) $B \circ_P X \to Y$. I think this is roughly what Garner does as well.

   I’m here. It seems Richard Garner got farther with this implementing this idea than I did, but Urs has the right spirit. The idea is to use a free contractible $P$-bimodule $B$, and define a weak functor $X \to Y$ to be a strict functor (a strict $P$-algebra map) $B \circ_P X \to Y$. I think this is roughly what Garner does as well.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeOct 21st 2010
   • (edited Oct 21st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexBy the way, maybe usful for Harry: back when I looked into this, the following little exercise made it all become very clear: construct explicitly for the [[folk model structure]] on strict 2-categories the cofibrant replacement of a strict 2-category $C$ and show that strict functors out of that are indeed "pseudofunctors" i.e. weak 2-functors. This is a tractable very explicit computation that illustrtates well the general mechanism at work here (the cofibrant replacement throws in precisely the cells whose image under a strict 2-functor become the "compositors" of the corresponding weak 2-functor).

   By the way, maybe usful for Harry:

   back when I looked into this, the following little exercise made it all become very clear:

   construct explicitly for the folk model structure on strict 2-categories the cofibrant replacement of a strict 2-category $C$ and show that strict functors out of that are indeed “pseudofunctors” i.e. weak 2-functors.

   This is a tractable very explicit computation that illustrtates well the general mechanism at work here (the cofibrant replacement throws in precisely the cells whose image under a strict 2-functor become the “compositors” of the corresponding weak 2-functor).

10. • CommentRowNumber10.
    • CommentAuthorHarry Gindi
    • CommentTimeOct 21st 2010
    • (edited Oct 21st 2010)
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    Author: Harry Gindi
    Format: MarkdownItexNm.

    Nm.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexHarry wrote: > Nm. I don't know what you want to say by this. But I made a note on the solution of the exercise that I mentioned <a href="http://ncatlab.org/nlab/show/2-category#FreeResolution">here</a>.

    Harry wrote:

    Nm.

    I don’t know what you want to say by this.

    But I made a note on the solution of the exercise that I mentioned here.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 25th 2010
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    Author: Todd_Trimble
    Format: MarkdownItexI think he means "never mind". But it would be nice to spell this out (literally!) than have to have a discussion about it.

    I think he means “never mind”. But it would be nice to spell this out (literally!) than have to have a discussion about it.

13. • CommentRowNumber13.
    • CommentAuthorHarry Gindi
    • CommentTimeOct 25th 2010
    • PermaLink
    Author: Harry Gindi
    Format: MarkdownItexYeah, I wrote something dumb, so I edited it and replaced it with the universal symbol for, "I'd better delete this before someone notices it".

    Yeah, I wrote something dumb, so I edited it and replaced it with the universal symbol for, “I’d better delete this before someone notices it”.

14. • CommentRowNumber14.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 5th 2017
    • (edited Sep 5th 2017)
    • PermaLink
    Author: Harry Gindi
    Format: MarkdownItexHey, the link died to this paper by Garner. Anyone remember what it was titled? Also, did anyone work out how to demonstrate that composition will be coherently associative in any way?

    Hey, the link died to this paper by Garner. Anyone remember what it was titled?

    Also, did anyone work out how to demonstrate that composition will be coherently associative in any way?

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeSep 5th 2017
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex[Homomorphisms of higher categories](http://comp.mq.edu.au/~rgarner/Papers/Womaps.pdf). He uses an algebraic cofibrant replacement to obtain a composition that's strictly associative.

    Homomorphisms of higher categories. He uses an algebraic cofibrant replacement to obtain a composition that’s strictly associative.

16. • CommentRowNumber16.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 6th 2017
    • (edited Sep 6th 2017)
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    Author: Harry Gindi
    Format: MarkdownItexNeat! Are there any results about co-Kleisli categories, for example, are they complete/cocomplete, is there anything about density (for example, if Q is a comonad on C and D c C is a full dense subcategory, and the inclusion of D in the co-Kleisli category is fully faithful, is D dense in the co-Kleisli category?), etc.? It sorta seems like co-Kleisli categories could be really badly behaved for a model of higher category theory (for example, failure of limits or colimits to exist, etc). Is there any reason to believe these co-Kleisli categories are well-behaved as places to do category theory?

    Neat! Are there any results about co-Kleisli categories, for example, are they complete/cocomplete, is there anything about density (for example, if Q is a comonad on C and D c C is a full dense subcategory, and the inclusion of D in the co-Kleisli category is fully faithful, is D dense in the co-Kleisli category?), etc.?

    It sorta seems like co-Kleisli categories could be really badly behaved for a model of higher category theory (for example, failure of limits or colimits to exist, etc). Is there any reason to believe these co-Kleisli categories are well-behaved as places to do category theory?

17. • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeSep 6th 2017
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    Author: Mike Shulman
    Format: MarkdownItex(co-)Kleisli categories are rarely complete or cocomplete. You can see this by noting that the Kleisli category of a monad is equivalent to the full subcategory of the Eilenberg-Moore category determined by the free objects. But there's no reason to expect a 1-category whose objects are higher categories to be well-behaved as a 1-category; it's remarkable enough that you can define a strictly associative composition on it in the first place. If what you're after is a model-category-like structure on the 1-category, then you generally need to either use a nonalgebraic model or use strict morphisms; Garner's comonad is then (morally) the cofibrant replacement in such a model structure.

    (co-)Kleisli categories are rarely complete or cocomplete. You can see this by noting that the Kleisli category of a monad is equivalent to the full subcategory of the Eilenberg-Moore category determined by the free objects. But there’s no reason to expect a 1-category whose objects are higher categories to be well-behaved as a 1-category; it’s remarkable enough that you can define a strictly associative composition on it in the first place. If what you’re after is a model-category-like structure on the 1-category, then you generally need to either use a nonalgebraic model or use strict morphisms; Garner’s comonad is then (morally) the cofibrant replacement in such a model structure.

18. • CommentRowNumber18.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 6th 2017
    • (edited Sep 6th 2017)
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    Author: Harry Gindi
    Format: MarkdownItexAh, as I suspected (I was thinking exactly of the fact that it's a subcategory of the Eilenberg-Moore category as a reason why it would be badly behaved)! It would all be too easy if it were well-behaved!

    Ah, as I suspected (I was thinking exactly of the fact that it’s a subcategory of the Eilenberg-Moore category as a reason why it would be badly behaved)! It would all be too easy if it were well-behaved!