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• CommentRowNumber1.
• CommentAuthorKeithEPeterson
• CommentTimeDec 29th 2016
• (edited Jun 21st 2018)

What’s to stop me from applying the ideas of non-standard analysis to category theory to generate $\omega$-categories that are ordered?

For instance, the vertical categorification n-transfor can be viewed as an action of the set $(\mathbb{N},0,+)$, given by the monoid action:

$nCat:\mathbb{N} \times Cat(C) \rightarrow \underset{n \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ C ,$

with identity given in the obvious way, this forms a category.

In this new category, we can now define new vertical categorification n-transfors by composing vertical categorification n-transfors, which can be given by the action:

$(n+m)Cat:(\mathbb{N} \times \mathbb{N}) \times Cat(C)$ $\rightarrow \underset{n \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ\underset{m \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ C.$

Now consider the vertical categorification $\omega$-transfors that is given by the infinite action of an ultrafilter on the vertical categorification n-transfor,

then the standard vertical categorification $\omega$-transfor is given by $\omega Cat(C) = [1+1+1+\cdots]Cat(C)$.

By applying the transfer principle from non-standard analysis, we can define some simple non-standard $\omega$-categories as such,

$(n_1+n_2+n_3+\cdots)Cat(C):[n_1,n_2,n_3,\cdots] \times Cat(C) \rightarrow \\\omega Cat(C) = [1+1+1+\cdots]Cat(C),$ $2\omega Cat(C) = [2+2+2+\cdots]Cat(C),$ $3\omega Cat(C) = [3+3+3+\cdots]Cat(C),$ $\cdots,$ $n\omega Cat(C) = [n+n+n+\cdots]Cat(C),$

some not so simple $\omega$-categories,

$\mathbb{N}^+\omega Cat(C) = [1+2+3+\cdots]Cat(C),$ $odd^+\omega Cat(C) = [1+3+5+\cdots]Cat(C),$ $Prime^+\omega Cat(C) = [2+3+5+7+\cdots]Cat(C),$ $\cdots,$

and also, regular vertical categorification n-transfors,

$Cat(C) = [1+0+0+\cdots]Cat(C),$ $2Cat(C) = [2+0+0+\cdots]Cat(C),$ $\cdots$ $nCat(C) = [n+0+0+\cdots]Cat(C),$

Is this line of reasoning valid, maybe it needs cleaning up? I’m curious what you all think.

P.S: What would I call these btw? “Hypercategory” is already taken.

• CommentRowNumber2.
• CommentAuthorKeithEPeterson
• CommentTimeDec 29th 2016
• (edited Jun 21st 2018)

By repeated application of ultraproducts above,

$[n_1+n_2+n_3+\cdots]Cat(C):[n_1,n_2,n_3,\cdots] \times Cat(C)$ $\rightarrow \underset{n_1 \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ\underset{n_2 \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ\underset{n_3 \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ\cdots\circ C,$ $[[n_1+n_2+n_3+\cdots]+[m_1+m_2+m_3+\cdots]+[l_1+l_2+l_3+\cdots],\cdots]Cat(C):$ $[[n_1,n_2,n_3,\cdots],[m_1,m_2,m_3,\cdots],[l_1,l_2,l_3,\cdots],\cdots] \times Cat(C)$ $\rightarrow \underset{[n_1+n_2+n_3+\cdots] \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ\underset{[m_1+m_2+m_3+\cdots] \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ\underset{[l_1+l_2+l_3+\cdots] \; times}{\underbrace{Cat\circ Cat\circ Cat\circ \cdots Cat}}\circ\cdots\circ C,$ $\cdots, etc,$

then there should exist a tower of polynomial enriched $\omega$-categories, which are given by the form:

$( x_{1} \cdot \omega^{n} +x_{2}\cdot\omega^{n-1}+\cdots+x_{n-2}\cdot\omega^{2}+x_{n-1}\cdot\omega+x_n )-Cat$
• CommentRowNumber3.
• CommentAuthorTobyBartels
• CommentTimeDec 30th 2016

First, you have to make sure that you're in a context in which the monoid action that you started with is well defined; when we're motivating higher categories (especially weak ones), this is more of a guideline than a definition. Still, there are now well established contexts in which this action is well defined, so we can suppose that you're working in one of those.

Then, the $\omega$ in nonstandard mathematics is not exactly the $\omega$ in $\omega$-category. (From now on, I will use ‘$\infty$’ for the latter, so ‘$\infty$-category’ instead of ‘$\omega$-category’.) In nonstandard analysis, $\lim_{n\to\infty} x_n$ is the standard part of $x_\omega$ (assuming that the latter has a standard part for every infinite integer $\omega$, and every infinite integer gives the same standard part), so I guess that you're suggesting that $\infty\Cat$ is the standard part of $\omega\Cat$ (and that the latter has a standard part for every infinite integer $\omega$, and every infinite integer gives the same standard part). That sounds reasonable and might be a useful perspective.

I don't know how much it matters to look at things like $x_{1} \cdot \omega^{n} +x_{2}\cdot\omega^{n-1}+\cdots+x_{n-2}\cdot\omega^{2}+x_{n-1}\cdot\omega+x_n$. As far as I know, that doesn't come up much in nonstandard analysis, so I wouldn't expect it to be useful here. But one can in principle talk about such things.

PS: I'd probably call these nonstandard higher categories.

• CommentRowNumber4.
• CommentAuthorKeithEPeterson
• CommentTimeDec 30th 2016
• (edited Jun 21st 2018)

I really need to get into a local course somewhere to help clean up my thoughts… >_>

Edit: I want want (strict) $\omega$-categories, because later on I’d like to be able to deal with vertical decategorification, treating the space as a groupiod. Keeping everything strict will keep things nice down the road for when I define the groupoid identity out of $(Decat \circ Cat)(C)$ in a way that preserves as much structure in sight. Later I want to Identitify decategorization as an action of truncated subtraction on this space. From there, I’d like to “non-stardardificate” it.

That’s my plan of action.

Edit edit: I think the viewing n-$cats$ as standard parts idea is interesting. Though, one would think a standard part $\infinity$-functor would work to cut out nCats, seeing as from the view of infinite ordinals, finite ordinals look like infinitesimal ordinals viewed from the vantage of finite ordinals in the nonstandard approach. Neat!

• CommentRowNumber5.
• CommentAuthorKeithEPeterson
• CommentTimeDec 30th 2016

I’m curious what the cardinality of such objects would even be, even in some of the simple cases.

• CommentRowNumber6.
• CommentAuthorKeithEPeterson
• CommentTimeJan 1st 2017

Actually, I wonder if the Baez-Dolan stabilization hypothesis still applies for nonstandard categories.

• CommentRowNumber7.
• CommentAuthorTobyBartels
• CommentTimeJan 1st 2017

It should, by the transfer principle. If $n$ is an integer, then an $(n+2)$-tuply monoidal $n$-category is stably monoidal; if this is true for whenever $n$ is a standard natural number, then it should remain true when $n$ is an infinite nonstandard natural number.

• CommentRowNumber8.
• CommentAuthorKeithEPeterson
• CommentTimeJan 6th 2017

Wait, how would I find the cardinality of a nonstandard $\infinity$-groupoid?

Perhaps,

$[n+n+n+\cdots]Grpd = \sum_{x \in \pi_0(X)}\prod_{j=1}^\infty {|\pi_{j\cdot n}(X,x)|}^{(-1)^{j\cdot n}}$

maybe?