$\newcommand{\cgh}{\mathsf{CGHaus}}$I'm reading through Goerss/Jardine's simplicial homotopy theory, and came across the word 'cofibration' for the first time. I tried to do some reading from other sources, but Goerss/Jardine seem to be using an ostensibly 'different' notion of cofibration to the rest of the world. I use notation standard to simplicial sets throughout.
I'll be specific now. Let $\cgh$ be the category of compactly generated Hausdorff spaces. The authors were discussing the 'closed model structure' of $\cgh$, wherein a fibration is defined to be a Serre fibration:
A (Serre) fibration is a map $f\in\cgh(U,V)$ such that for all integer $n>0$ and all $0\le k\le n$, $f$ has the right lifting property against the inclusion $|\Lambda^n_k|\hookrightarrow|\Delta^n|$.
I'm fairly sure this definition is equivalent to the (seemingly) more standard definition of the same in terms of lifting against $|\Delta^n|\hookrightarrow|\Delta^n|\times|\Delta^1|$.
They define:
A cofibration $i\in\cgh(U,V)$ is a map that has the left lifting property against all trivial fibrations.
A 'trivial' fibration is a fibration that is also a weak homotopy equivalence. This definition seems very removed from the definitions of the same on nLab: I could hardly even parse those!
My first question:
- Is the Goerss-Jardine definition equivalent to the standard meaning of cofibration (whatever that is?)?
On another note, they proved (for simplicial fibrations) that a map is a trivial fibration iff. it has the right lifting property against all inclusions $\partial\Delta^n\hookrightarrow\Delta^n,n\ge0$. According to nLab, this holds in the topological world - a map is a Serre fibration iff. it has the lifting property against the inclusions $S^{n-1}\hookrightarrow D^n,n\ge0$.
- Is there a similar characterisation of the trivial cofibrations? (in either the 'standard' sense or in the sense of Goerss-Jardine?)
The answer to your question is yes.
The general setup is as follows: in order to do homotopy theory, the basic framework is a model category (and we can focus on the two examples of simplicial sets and an appropriate category of topological spaces, like $\mathsf{CGHaus}$). To define a model category, we need three classes of maps:
As you say, the "trivial fibrations" are the maps that are both fibrations and weak equivalences, and similarly for the trivial cofibrations.
The weak equivalences are the maps we want to invert, and we typically know those ahead of time: they are weak homotopy equivalences in our two examples. However we define the other two, we need the cofibrations to have the left lifting property with respect to trivial fibrations, and we need the fibrations to have the right lifting property with respect to trivial cofibrations.
Of the other two, sometimes it's easier to define cofibrations, sometimes fibrations. For simplicial sets, it is easiest to define cofibrations — they're just inclusions. For topological spaces, it is easiest to first define some maps that you want to be trivial cofibrations (the inclusions $i_0: D^n \to D^n \times I$) and then define the fibrations in terms of a lifting property with respect to those, and then define the cofibrations in terms of a lifting property with respect to the trivial fibrations. You can choose different families for these "generating cofibrations," and the ones that Goerss and Jardine use are just as good as $D^n \to D^n \times I$. With some work, you can then say that certain families of maps, like the inclusions of subcomplexes into CW complexes, are cofibrations. I don't know how easy it is to give a self-contained description of all cofibrations, though. (By "self-contained," I mean without reference to the fibrations.)
(To make the "generating cofibration" approach work, you should also define some maps that you know that you want to be cofibrations. A good choice is the family $S^{n-1} \to D^n$, also known as $|\partial \Delta^n| \to |\Delta^n|$. See Lemma 9.4 in Goerss-Jardine, for example.)