First steps in Higher Topos Theory 2

To understand Higher Topos Theory it is important to master (among other things) the basics of the following subjects in category theory:

1. Monoidal categories

2. Enriched categories (including ends and coends and enriched Yoneda lemma)

3. Model categories (including combinatorial model categories and homotopy (co)limits)  

For 1 and 2 a good reference is Part I of Birgit Richter's From Categories to Homotopy Theory.

For 3 it is the book chapter Homotopy theories and model categories by W. G. Dwyer and J. Spalinski.

But there is more. One must feel very at home with simplicial sets (and their connection to homotopy types) which play a central role in Higher Topos Theory.  Finally one must master the basics of Grothendieck toposes.

Instead of ordinary (set-enriched) model categories we work with model categories enriched over simplicial sets sSet (seen itself as a category enriched over simplicial sets and when endowed with its classical (Quillen) model structure its fibrant-cofibrant objects are called $\infty$-groupoids).  Instead of presheaves to sets we work with simplicial set enriched functors from the opposite of a simplicial set enriched category to simplicial sets. A key result is Dugger's theorem, the simplicial presheaf analogue for combinatorial model categories of the characterization of sheaf toposes as left exact full subcategory localizations of presheaves - itself a generalization of the presentation of an object by generators and relations. 

So Higher Topos Theory is Topos Theory done (homotopically) over sSet rather than Set.  

It seems we can give a more geometric interpretation of the nerve of category $C$ (a canonical way of extracting an simplicial set ) given usually  in terms of composable sequences of arrows. For take three composable arrows $f,g,h$. Think of $f$ and $g$ as being in the plane but $h$ directed perpendicularly into space. Then we get 3-simplex in $N(C)_3$ with faces $(f,g, g\circ f)$, $(g,h, h\circ g)$ and $(g\circ f, h, h\circ (g \circ f))$ where we view a pair of composable arrows together with their composition as a triangle, i.e. a 2-simplex. Our intuition is that the nerve of a category keeps track of all commutative diagrams and each such diagram is a geometric object.

Of course there are competing definitions of (models of) $\infty$-groupoids besides the sSet-based one (Kan complexes) which itself is only one possible choice for shapes (which include cubical and cellular sets).

All this suggests philosophically that our different concepts and models of what a 'space' is are special embodiments of a single 'pure' concept which is yet to be determined. Note that homotopy type theory connects $\infty$-groupoids to  (identity) types/ propositions or spaces of proofs/functions/computations.