Why is the class of isomorphisms perfect? Lurie, p.824

Question

How does one prove that example in pg.824, A.2.6.11, satisfies axiom (4) ?

We want to show that for $C$ a presentable category, $W$ it sclass of isomorphism,

There exists a small set $W_0\subseteq W$ such that every morphism to $W$ can be obtained as a filtered colimit of morphisms belong to $W_0$.

It is not clear to me how to pick $W_0$ - with the axioms of presentable category, in A.1.1.1.

Answer 1

I will use the more standardised terminology for this setting, see for example Locally Presentable and Accessible Categories by Adámek and Rosický. I will put the corresponding terminology from Lurie's document you linked behind it in brackets.

The definition from Lurie of a presentable category is also nonstandard. The usual definition is as follows:

A category $\mathcal{C}$ is called a locally presentable category (presentable category) if the following hold:

1. $\mathcal{C}$ is cocomplete,
2. there is a small set $\mathcal{A}$ of $\lambda$-presentable objects ($\lambda$-compact objects), such that every object in $\mathcal{C}$ is a $\lambda$-filtered colimit of a diagram with objects from $\mathcal{A}$.

Furthermore, I think that when Lurie says "filtered colimit" he means "$\lambda$-filtered for some $\lambda$", because in a lot of other contexts "filtered colimit" means "$\omega$-filtered colimit".

The reason this is relevant for your question is because this way it will pretty much follow from the definition. Let $\mathcal{A}$ be the set as in the definition above, then we can take $W_0$ to be the set of all identity arrows on objects in $\mathcal{A}$. Let $f: X \to Y$ be any isomorphism. We have that $X$ is the colimit of some $\lambda$-filtered diagram $(A_i)_{i \in I}$ in $\mathcal{A}$. Let $(p_i: A_i \to X)_{i \in I}$ denote the coprojections. Then $(fp_i: A_i \to Y)_{i \in I}$ form a cocone, which is also colimiting since $f$ is an isomorphism. So $f: X \to Y$ is the induced arrow between the colimits of $(A_i)_{i \in I}$ and, well, $(A_i)_{i \in I}$. In other words, $f$ is the $\lambda$-filtered colimit of $(Id_{A_i})_{i \in I}$.