# Dugger and Spivak's combinatorial proof of Joyal's isofibration theorem, a fortiori?

In what follows, let $$E^n = \operatorname{cosk}_0(\Delta^n)$$

Joyal's isofibration theorem says precisely

An inner fibration $$p:X\to Y$$ of quasi-categories is a fibration for the Joyal model structure if and only if it has the right lifting property with respect to the vertex inclusion $$e: \Delta^0=E^0 \hookrightarrow E^1$$

An immmediate consequence of this theorem is that the fibrant objects in the Joyal model structure are precisely the quasi-categories.

In their paper Mapping Spaces in Quasi-Categories, Dugger and Spivak gave an alternative direct combinatorial argument of one of the lemmas in Joyal's paper, Lemma A.4.

The statement of this lemma is

For $$n>0$$, the corner product of $$b^n:\partial \Delta^n \hookrightarrow \Delta^n$$ with $$e$$, $$e\times^\lrcorner b^n:\Delta^0 \times \Delta^n\coprod_{\Delta^0 \times \partial \Delta^n}E^1 \times \partial \Delta^n\hookrightarrow E^1 \times \Delta^n$$ is special anodyne.

The definition of special anodyne here is not important, because this is the question: When reading through the proof of this statement, they only directly prove that the inclusion of stage $$m$$ of the filtration into stage $$m+1$$, written as $$Y_m \hookrightarrow Y_{m+1}$$, is inner anodyne for $$0 < m < n,$$ leaving the proof in the 'special' outer cases to the reader by just outlining the argument.

However, upon re-analyzing their argument myself, I found no reason why the maps $$Y_0 \hookrightarrow Y_1$$ and $$Y_n \hookrightarrow Y_{n+1}$$ are not also inner anodyne. Those two cases appear to be precisely identical as far as the structure of the proof goes.

It's now known by a completely different proof of Danny Stevenson (see Example 5.8) that in fact, the maps $$e\times^\lrcorner b^n$$ are all inner anodyne (not merely special anodyne) for $$n>0$$.

I contacted Dan Dugger and David Spivak by E-mail to ask them if in fact they had indeed proven the stronger statement:

For $$n>0$$, the corner product of $$b^n:\partial \Delta^n \hookrightarrow \Delta^n$$ with $$e$$, $$e\times^\lrcorner b^n:\Delta^0 \times \Delta^n\coprod_{\Delta^0 \times \partial \Delta^n}E^1 \times \partial \Delta^n\hookrightarrow E^1 \times \Delta^n$$ is inner anodyne,

but neither of them could remember the argument very well. When I suggested to David that perhaps they weakened their claim so that it would agree with the statement known at the time due to Joyal, he said that he had fuzzy memories of maybe doing something like this, but he wasn't at all sure.

He suggested I e-mail another mathematician closely acquainted with this combinatorial argument, but she also had forgotten the details of this rather technical combinatorial lemma.

So I ask, did they indeed prove the stronger claim? I have a use for this, since it actually exhibits $$e\times^\lrcorner b^n$$ with an even stronger property than that proved by Danny Stevenson, namely that it is a relative cell complex for the inner horns.

• For those wondering, Joyal's isofibration theorem follows immediately from this stronger claim, since using Cisinski's theory together with the fact that inner anodynes are closed under corner with monomorphisms, we see that the fibrant objects are immediately the quasicategories. Since the naïve fibrations between fibrant objects are exactly the fibrations between them, the full statement then follows immediately, since for an inner fibration to be a naïve fibration, it need only have the right lifting property with respect to $e$, since lifts exist for all other generating anodynes. – Harry Gindi Dec 4 at 9:46
• I think that in order to follow precisely their argument you need the special outer horns. Consider $n=1$ (or $r=1$ in the notation of their paper). In order to extend $Y^1[1]$ to $Y^1[2]$ you need for instance to add the 1-simplex $([a_0], [b_1])$ that you consider as a face of the 2-simplex $([a_0], [b_1, a_1])$. But this is an outer horn of the form $\Lambda^2_2 \to \Delta^2$ and the face $(\emptyset, [b_1, a_1])$ is an equivalence, so a special right horn. – Andrea Gagna Dec 4 at 16:23
• Nevertheless, I agree with you: it seems to me that modifying just a bit the procedure of the last case (i.e. moving the added $a_r$ of the simplexes they call $x'$ at the beginning of the $r$-group), then also the last inclusion $Y^r \to Y^{r+1} = Y$ is an inner anodyne extension. – Andrea Gagna Dec 4 at 16:26
• @AndreaGagna Yeah, I was thinking this as well, but it got really messy and I wasn't sure if it was working out correctly. – Harry Gindi Dec 4 at 17:56