I am reading Cisinski's Higher Categories and Homotopical Algebra and I am having trouble trying to verify some claims there. My background in category theory is not very solid. I would like some help to understand and prove one of that claims.
Let $i: A \to B$ and $p: X \to Y$ be two morphisms in a category $\mathcal{C}$. We say that $i$ has the left lifting property with respect to $p$, or, equivalently, that $p$ has the right lifting property with respect to $i$, if any commutative square of the form
has a diagonal filler
(i.e. a morphism $h$ such that $hi = a$ and $ph = b$).
A class of morphisms $F$, in a category $\mathcal{C}$, is stable under transfinite compositions if, for any well ordered set $I$, with initial element $0$, for any functor $X: I \to \mathcal{C}$ such that, for any element $i \in I$, $i \neq 0$, the colimit $\varinjlim_{j < i} X(j)$ is representable and the induced map
$$\varinjlim_{j < i} X(j) \to X(i)$$ belongs to the class $F$, the colimit $\varinjlim_{i \in I} X(i)$ exists and the canonical morphism $X(0) \to \varinjlim_{i \in I} X(i)$ belongs to $F$ as well.Proposition: Let $\mathcal{C}$ be a category with a class of morphisms $F$. Then the class $l(F)$ is stable under transfinite compositions.
The above proposition is just a small part of the Proposition 2.1.4 in the book. I was able to prove the other parts, so I am not completely lost. The problem is in this definition of transfinite composition... I know what a colimit is, I know what a well ordered set is, but I am not sure I understood the definition at all. Where does the condition "the colimit $\varinjlim_{j < i} X(j)$ is representable" comes into play? What is the canonical morphism $X(0) \to \varinjlim_{i \in I} X(i)$? Is this one of that morphisms in the definition of cocone? How does the above definition is equivallent to the nLab definition of transfinite composition? How to prove the above Proposition?
Cisinski means that the colimit exists, in more common terminology for English-speaking mathematicians. (In the Grothendieck school, it is traditional to speak of a colimit as a presheaf, so that it always exists, and the question is whether this presheaf is representable in the starting category.)
Now, as for what the condition says: first, consider the case that $i=i'+1$ is a successor ordinal. Then $\mathrm{colim}_{j<i}X(j)=X(i'),$ and so the condition simply says that the map $X(i')\to X(i)$ is in $F.$ The case in which $i$ is a limit ordinal is the natural way to say what is morally the same thing: for instance, the map $\mathrm{colim}_{i\in\omega} X(i)\to X(\omega)$ is in $F.$ If you're homotopy-theoretically minded you might picture this as the union of a countable chain of cofibrations, all mapping into $X(\omega).$
Yes, the canonical morphism from $X(0)$ to the colimit must be the appropriate leg of the colimiting cocone.
With all this spelled out, the proposition should be easy to prove by transfinite induction.