Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Little page to focus on this important notion, as opposed to the general remarks at walking structure.
This should be tied in or amalgamated with interval groupoid, which is a redirect for (and section of) interval category.
Yes, I agree. I will attempt something now, and then others can refine if needed.
Adding redirect for ’interval groupoid’, and adding remark that the walking isomorphism is the free groupoid on the walking arrow, which was previously at interval category.
Noting various 2-categories which categorify it. I will add the walking 2-isomorphism later, which I am intending to use at Lack fibration; run out of time for the moment!
Actually what I wish to use at Lack fibration is something with trivial boundary, i.e. only one object and only identity 1-arrows, and exactly two non-identity 2-arrows which are the inverses of a 2-isomorphism (I think the horizontal composites are dictated by the interchange law to be the identity then). This is not really the walking 2-isomorphism, in that it does not represent 2-isomorphisms. Let me know if you have a suggestion for alternative terminology!
I have a question about the nerve of the walking isomorphism.
Let $J(n) \subseteq N(J)$ be the subsimplicial set that is the image of the nondegenerate $n$-simplex starting from vertex $0$.
I’m pretty sure that for $n \geq 3$, $J(n) \to N(J)$ is supposed to be a categorical equivalence of simplicial sets (i.e. a weak equivalence in the Joyal model structure for quasi-categories), and I know that the inclusion $J(n) \to J(n+1)$ is left anodyne, being the pushout of $\Lambda^{n+1}_0 \to \Delta^{n+1}$. (and therefore $J(3) \to J$ is left anodyne)
Are the inclusions $J(n) \to J(n+1)$ inner anodyne? I thought I’ve worked out once that they are but I can’t reconstruct the argument, and the only references I can find at the moment on related issues are in terms of an analysis of the construction via left anodyne maps.
I can’t see how to prove that $J(n) \to J(n+1)$ is inner anodyne. However, observe that, if $n \geq 3$, it does have the left lifting property with respect to all inner fibrations between quasi-categories, by Joyal’s lifting theorem. Since $J$ is a quasi-category, it follows that $J(n) \to J$ is inner anodyne, for $n \geq 3$.
Now that I’ve written out my question, I (think I) was able to remember my argument, and where my memory about the conclusion went wrong. Basically, it involved taking the pushout along a monomorphism of a splitting of an inner anodyne map, so I was only constructing trivial cofibrations in the Joyal model structure, not inner anodyne maps.
@Richard #7
Actually what I wish to use at Lack fibration is something with trivial boundary, i.e. only one object and only identity 1-arrows, and exactly two non-identity 2-arrows which are the inverses of a 2-isomorphism
Isn’t this $\mathbf{B}^2\mathbb{Z}/3$? It has a unique object, a unique 1-arrow, and three 2-arrows. You can think of the latter as being $\{-1,0,+1\}$ under addition, if you like.
1 to 13 of 13