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It is a theorem of Joyal that the class of all anodyne maps in the Joyal model structure is generated by (equivalently):
Inner horn inclusions
or
$\Delta^n\times \Lambda^2_1 \cup \partial\Delta^n \times \Delta^2\to \Delta^2 \times \Delta^n$or
$S' \times \Lambda^2_1 \cup S \times \Delta^2\to \Delta^2\times S'$for all inclusions $S\hookrightarrow S'$.
Now, if we use Cisinski’s formalism of classes of anodyne maps, the equivalence of the second two become obvious (by paragraph 1.3.12 and remark 1.3.15) if we can figure out what the cylinder functor for the Joyal model structure is. Now, for sure we know that the actual cylinder functor is taking the product with $\Delta^2$, but we need slightly more information regarding the actual structure of the cylinder. Obviously, the retraction of the two maps $X\to \Delta^2\times X$ will be the projection on the second factor, but I can’t figure out for the life of me what the two maps $X\to \Delta^2\times X$ actually are. It seems to be related to the inclusion of the inner 2-horn, but I can’t get any further.
Maybe you guys can figure out what the situation is? I’m inclined to believe that the maps are the inclusions of the legs of the inner 2-horn, where the inner 2-horn takes the role of $X\coprod X = \partial I \times X$.
Do you mean inner anodyne maps? I didn’t think the class of all acyclic cofibrations in the Joyal model structure had any explicit generating set.
I meant what I said, although Cisinski’s terminology conflicts with the terminology of Lurie and Joyal. A class of anodyne maps for a cylinder means something rather specific.
What I was asking for was finding a cylinder functor generating the class of weak cofibrations in the Joyal model structure (i.e. inner anodyne maps) by means of the empty generating set for the anodyne structure for that cylinder object. I don’t have time at the moment, but I will translate all of the relevant material if you don’t feel like reading it yourself.
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