# Proposition A.2.6.15 of Higher Topos Theory

I am reading Higher Topos Theory to learn the properties of $$\infty$$-categories. I found that Proposition A.2.6.15 is used in Proposition 2.1.4.7 to give a construction of the covariant model structure. But Proposition A.2.6.15 uses some features that had not been introduced until Chapter 5, making its proof inaccessible before reading Chapter 5.

Is there any proof that avoids using those structures that are introduced later in this book?

• Quite possibly. People have improved on the proofs of HTT somewhat here and there. There is also a proof in mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf but I don't know how self-contained it is. You may also find math.univ-lille1.fr/~nwejm/OnlinePapers/Archives/2017/3/7/… useful. – David Roberts Mar 7 at 3:24
• Cisinski also discusses the covariant model structure in his recent book, "Higher categories and homotopical algebra". – Charles Rezk Mar 7 at 3:28
• It should be pointed out that the material from Ch 5 used in that appendix is just the special case of presentable and accessible $\infty$-categories applied to ordinary $1$-categories. In that form, the theory is well developed, see e.g. the book by Adamek and Rosicki. "Presentable categories" are usually called "locally presentable categories" in that context. – Charles Rezk Mar 7 at 3:30
• Thanks for your help! I will try these materials to get further access. – Frank Kong Mar 7 at 3:47