Basic Homotopy Theory
Table of Contents
This node discusses basic homotopy theory (by this we mean a category equipped with weak equivalences), its corresponding notions and how it relates to \( \infty \)-categories.
A category with weak equivalences consists of
- a category \( \mathcal C \),
- a class \(W \subseteq \Mor \mathcal C \) (of weak equivalences),
such that
- \(W \) contains all isomorphisms,
- \(W \) satisfies the 2-out-of-3 property.
Let \( (\mathcal C , W)\) be a category with weak equivalences. The associated homotopy category is the localization \(\mathcal C[W ^{-1}] \).
1. TODO Homotopy Limits and Colimits
2. Associated \(\infty \)-Categories
When we go from a category \((\mathcal C,W) \) with weak equivalences to its homotopy category, we lose a lot of information. Maybe we should think of a reason why?? For example there are “non-equivalent” (what ever that means) categories with weak equivalences such that the underlying homotopy categories are equivalent. Hence we can ask for a finer homotopy category, a step between \((\mathcal C,W) \) and its homotopy category. This is were \(\infty \)-categories come into place.
Let \((\mathcal C,W) \) be a category with weak equivalences. The associated \(\infty \)-category is \(N(\mathcal C)[W ^{-1}] \).
The next lemma confirms that the associated \(\infty \)-category is indeed a step between \((\mathcal C,W) \) and its homotopy category.
Let \((\mathcal C,W) \) be a category with weak equivalences. Then the homotopy category \(H(\mathcal C,W) \) of \((\mathcal C,W) \) is equivalent (as a \(1 \)-category) to the homotopy cateogry \(h(H _{\infty }(\mathcal C,W) )\) of the associated \(\infty \)-category.
We suspect that the following theorem (or a version of it is true), but don’t have a reference or a proof. Our conjecture comes from the fact that nLab says that all simplicial categories are Dwyer-Kan equivalent to a simplicial (Hammock?) localization of a category with weak equivalences.
2.1. TODO Land Theorem 3.3.8!!
Every \(\infty \)-category \(\mathcal C \) is Joyal equivalent to an associated \(\infty \)-category of a category with weak equivalences.
2.2. Other stupid constructions that we don’t understand
Let \((\mathcal C,W) \) be a category with weak equivalences. The hammock localization of \((\mathcal C,W) \) is \(N(L^H(\mathcal C,W)) \) where \(L^H(\mathcal C,W) \) is the simplicial category??
Let \(U \colon \Cat \rightarrow \mathrm{Grph} \) be the forgetful functor to the categories of reflexive graphs. Let \(F \colon \mathrm{Grph} \rightarrow \Cat \) be the left adjoint of \(U \). Let \(T := FU \colon \Cat \rightarrow \Cat \) be the induced comonad. Let \(T _{\bullet } \colon \Cat \rightarrow [\Delta ^{op}, \Cat ]\) be the standard resolution (??) with respect to \(T \). Then the standard resolution of \(\mathcal C \) is \(T _{\bullet }\mathcal C \). Idk what to do from here (see nLab simplicial localization).
3. Model Categories
We first define what a model structure on a category (or a model category) is.
Let \(\mathcal C \) be a \(1 \)-category. A model structure on \(\mathcal C \) consists of
- a class \(\Cof \subseteq \Mor \mathcal C \) of cofibrations,
- a class \(\Fib \subseteq \Mor \mathcal C \) of fibrations,
- a class \(W \subseteq \Mor \mathcal C \) of weak equivalences,
such that
- \(W \) contains all isomorphisms,
- \(W \) satisfies the 3-out-of-2 property,
- every morphism can be factored into a trivial cofibration followed by a fibration,
- every morphism can be factored into a cofibration followed by a trivial fibration,
- \(\chi_L(W \cap \Fib ) = \Cof \) and \(\chi_R(\Cof ) = W \cap \Fib \),
- \(\chi_R(W \cap \Cof) = \Fib \) and \(\chi_L(\Fib ) = W \cap \Cof \).
A model category is a category \(\mathcal C \) that is complete and cocomplete and is equiped with a model structure on \(\mathcal C \).