When is it true that $\varinjlim_i[X,Y_i]\cong[X,\varinjlim_iY_i]$

Question

Let $X$ be a topological space and $(Y_i,f_{ji})_{i\in I}$ be a direct system. I believe that there is a natural map $$\varinjlim_i[X,Y_i]\to[X,\varinjlim_iY_i]$$, where $[X,Y_i]$ denotes the homotopy classes of maps between $X$ and $Y_i$.

My question is:

1. Are there some useful conditions on $X$ and $(Y_i,f_{ji})$ to make the above natural map bijective?
2. Besides, let $I=\mathbb{N}$. Assume $X$ is compact, each $f_{ji}$ is a closed embedding, and each $Y_i$ is paracompact. Is it the case that the mentioned natural map is bijective?

I am asking this question because sometimes I have to check that the mentioned natural map is bijective while I don't have some available useful results at hand. Therefore, I am trying to gather some useful results. Thanks.

Answer 1

Now, I can answer my second question. The key observation is the following

Let $Y=\cup_{i\geq0}Y_i$ be a $T_1$ space where $Y_i\subset Y_{i+1}$ is closed and $Y$ is equipped with the direct limit topology. Then, every compact set $C\subset Y$ is contained in some $Y_n$.

Assuming the contrary, for each $i\geq0$, we pick an element $x_i\in C\setminus Y_i$ and form an infinite sequence $\{x_i\}\subset C$. For any subset $A\subset\{x_i\}$, $A\cap Y_i$ is finite and hence closed in $Y_i$. Therefore, $A$ is closed in $Y$, so that $\{x_i\}$ is a discrete closed set in $Y$. Moreover, as a closed subset of $C$, $\{x_i\}$ is compact, so that it must be finite. This is a contradiction.

This observation clearly implies that $\varinjlim[X,Y_i]\to[X,Y]$ is surjective, since for any map $f\colon X\to Y$, $f(X)\subset Y_n$ for some $n$. Moreover, it is injective. In fact, if $f_i\colon X\to Y_i$ and $f_j\colon X\to Y_j$ are homotopic in $Y$, then there is a homotopy $H\colon X\times I\to Y$. Since $H(X\times I)\subset Y_n$ for some $n$, $f_i$ and $f_j$ are homotopic in $Y_n$. This means that they are equal in $\varinjlim[X,Y_i]$.

Answer 2

If $X$ is a finite-dimensional CW complex and $Y_i$ are CW complexes such that $Y_i$ is the $i$-skeleton of $Y := \lim_i Y_i$ and $\dim Y \geq \dim X + 1$, then the natural map is a bijection by the cellular approximation theorem. In fact, all you need is that $\dim Y_n \geq \dim X + 1$ for some $n$ and that for $m > n$, $Y_m$ is obtained from $Y_n$ by attaching cells of dimension greater than $\dim X + 1$.