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1. • CommentRowNumber1.
   • CommentAuthorbarron
   • CommentTimeOct 18th 2015
   • (edited Oct 18th 2015)
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   Author: barron
   Format: MarkdownItexRecently I was thinking about a certain construction. I will put it forth here in case it piques somebody else's interest like it piqued mine. I don't consider this to be a very serious topic so I speak in broad strokes meant to stimulate the intuition. (Furthermore I am no category theorist.) Let $\mathbf{Turtle}$ be the category with elements $\mathbb{Z}$ and arrows generated by $s_i, t_i: n \to n+1$ which satisfy $s_{i+1} \circ s_i = t_{i+1} \circ s_i$ and $s_{i+1} \circ t_i = t_{i+1} \circ t_i$ (cf. _globe category_). Now say a _turtular set_ is a presheaf on $\mathbf{Turtle}$ (cf. _globular set_). The reason for the nomenclature is that, while a preasheaf on the globe category has objects (the image of the object zero in the globe category), arrows between objects (image of 1), 2-arrows (image of 2), etc., i.e. it has all sorts of arrows but objects aren't arrows between anything; in the turtle category it's [turtles all the way down](https://en.wikipedia.org/wiki/Turtles_all_the_way_down) (it's just (n)-arrows between (n-1)-arrows, for all $n \in \mathbb{Z}$). This is a peculiar sort of generalization of a globular set, as given a turtular set $T$ we can approximate a globular set $G$ by making $T$ have only one $n$-arrow for all $n \lt 0$, and saying that the 0-arrows of $T$ are the objects of $G$, and the $n$-arrows of $G$ (for $n \in \mathbb{N}_{\gt 0}$) are precisely the $n$-arrows of $T$. In the same way that a globular set models a strict $\omega$-category, a turtular set models a strict ["infinitely deep"](http://math.stackexchange.com/questions/1479504/infinitely-deep-globular-sets-categories) category (the intuition is not super hard to grasp here, I think). If we now think of the Batanin $\omega$-category construction (an algebra over a globular operad; the result is a weak $\omega$-category, i.e. an $(\infty,1)$-category) and try to construct (in our heads; don't delve into the formalities) the analogous "thing" for turtular sets, we get this thing which is like a weak $\omega$-category, except, there are no "objects" which terminate the $\mathbb{Z}$-heirarchy of arrows, now the arrows go "all the way down". The "thing" is maximally weak in the sense that for every $n$, composition of $n$-arrows is not strictly unital and associative, only up to (presumably coherent) isomorphism. It is not unreasonable to call such a "thing" a _turtle category_ or perhaps just a $\mathbb{Z}$-category (cf. $\omega$-category := $\mathbb{N}$-category). (Note that if we were to truncate a turtle category at zero (by having only one $n$-arrow for $n \lt 0$), then we get something like a weakly monoidal $(\infty, 0)$-category.) One shortcoming: I can't find any example of this structure "in the wild". By which I mean an example which is not trivial and not arbitrarily constructed. (cf. for $(\infty, 1)$-categories an example "in the wild" is higher homotopy structures.) If someone can find a nice example of such a "turtle category" as described above, and is attending JMM 2016, I will buy you a cup of coffee (or if you prefer, a cannabis cigarette, as I hear that is legal in Seattle these days).

   Recently I was thinking about a certain construction. I will put it forth here in case it piques somebody else’s interest like it piqued mine. I don’t consider this to be a very serious topic so I speak in broad strokes meant to stimulate the intuition. (Furthermore I am no category theorist.)

   Let $\mathbf{Turtle}$ be the category with elements $\mathbb{Z}$ and arrows generated by $s_i, t_i: n \to n+1$ which satisfy $s_{i+1} \circ s_i = t_{i+1} \circ s_i$ and $s_{i+1} \circ t_i = t_{i+1} \circ t_i$ (cf. globe category). Now say a turtular set is a presheaf on $\mathbf{Turtle}$ (cf. globular set). The reason for the nomenclature is that, while a preasheaf on the globe category has objects (the image of the object zero in the globe category), arrows between objects (image of 1), 2-arrows (image of 2), etc., i.e. it has all sorts of arrows but objects aren’t arrows between anything; in the turtle category it’s turtles all the way down (it’s just (n)-arrows between (n-1)-arrows, for all $n \in \mathbb{Z}$). This is a peculiar sort of generalization of a globular set, as given a turtular set $T$ we can approximate a globular set $G$ by making $T$ have only one $n$-arrow for all $n \lt 0$, and saying that the 0-arrows of $T$ are the objects of $G$, and the $n$-arrows of $G$ (for $n \in \mathbb{N}_{\gt 0}$) are precisely the $n$-arrows of $T$. In the same way that a globular set models a strict $\omega$-category, a turtular set models a strict “infinitely deep” category (the intuition is not super hard to grasp here, I think).

   If we now think of the Batanin $\omega$-category construction (an algebra over a globular operad; the result is a weak $\omega$-category, i.e. an $(\infty,1)$-category) and try to construct (in our heads; don’t delve into the formalities) the analogous “thing” for turtular sets, we get this thing which is like a weak $\omega$-category, except, there are no “objects” which terminate the $\mathbb{Z}$-heirarchy of arrows, now the arrows go “all the way down”. The “thing” is maximally weak in the sense that for every $n$, composition of $n$-arrows is not strictly unital and associative, only up to (presumably coherent) isomorphism. It is not unreasonable to call such a “thing” a turtle category or perhaps just a $\mathbb{Z}$-category (cf. $\omega$-category := $\mathbb{N}$-category). (Note that if we were to truncate a turtle category at zero (by having only one $n$-arrow for $n \lt 0$), then we get something like a weakly monoidal $(\infty, 0)$-category.)

   One shortcoming: I can’t find any example of this structure “in the wild”. By which I mean an example which is not trivial and not arbitrarily constructed. (cf. for $(\infty, 1)$-categories an example “in the wild” is higher homotopy structures.)

   If someone can find a nice example of such a “turtle category” as described above, and is attending JMM 2016, I will buy you a cup of coffee (or if you prefer, a cannabis cigarette, as I hear that is legal in Seattle these days).

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 18th 2015
   • (edited Oct 18th 2015)
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   Author: DavidRoberts
   Format: MarkdownItexIn Baez and Dolan's _Categorification_, they talk about Z-categories and Z-groupoids, which is similar to what you are talking about. You should be able to construct examples of your gadgets from [[spectrum|spectra]]. I'd grab that coffee, but I'm not going to the US any time soon :-)

   In Baez and Dolan’s Categorification, they talk about Z-categories and Z-groupoids, which is similar to what you are talking about. You should be able to construct examples of your gadgets from spectra.

   I’d grab that coffee, but I’m not going to the US any time soon :-)

3. • CommentRowNumber3.
   • CommentAuthorKarol Szumiło
   • CommentTimeOct 18th 2015
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   Author: Karol Szumiło
   Format: MarkdownItexThis is only vaguely related, but there is an old paper by Kan called _Semisiplicial Spectra_. These semisimplicial spectra are a kind of simplicial version of your gadget with simplicial operators going all the way down. Kan shows that they model spectra. This suggest that it might be a good idea to look for examples in stable homotopy theory.

   This is only vaguely related, but there is an old paper by Kan called Semisiplicial Spectra. These semisimplicial spectra are a kind of simplicial version of your gadget with simplicial operators going all the way down. Kan shows that they model spectra. This suggest that it might be a good idea to look for examples in stable homotopy theory.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 18th 2015
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   Author: DavidRoberts
   Format: MarkdownItexTo be more concrete, given a chain complex (C,d) you can define a turtular set, by saying that for x in C(n), s(x)=0 and t(x)=d(x).

   To be more concrete, given a chain complex (C,d) you can define a turtular set, by saying that for x in C(n), s(x)=0 and t(x)=d(x).

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 19th 2015
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   Author: Urs
   Format: MarkdownItexre #3, see _[[combinatorial spectrum]]_

   re #3, see combinatorial spectrum