According to the paper “A theory of algebraic cocycles” (this is a summary of the original paper), theorem 8, gives the morphic cohomology of weight one. More precisely for any complex quasi-projective variety $X$ (the theorem mentions projective but it is expected to be true for quasi-projectives too), one has the following isomorphisms:

$$L^1H^0(X)cong mathbb{Z}, L^1H^1(X)cong H^1(X,mathbb{Z})$$

These two morphic cohomology groups can be identified with $pi_2 (coprod_n Mor(X, mathbb{P}^{n}))^+$ and $pi_1 (coprod_n Mor(X, mathbb{P}^{n}))^+$ respectively (you might also need to check the definition of morphic cohomology and also remark 3.6 in the original paper). The plus sign is the group completion of the monoid. Since these are supposed to be true for all quasi projective varieties one can expect that these isomorphisms hold for local ring and function fields by taking colimit. So I just want to verify this for a function field of a variety. Let’s call the field $F$. There is an easy way to describe $Mor(X, mathbb{P}^{n})$ for affine $X$ and especially for $X=text{Spec}(F)$. It amounts to giving a line bundle on $text{Spec}(F)$ which is trivial and elements in $s_1, cdots s_{n+1}$ in the field $F$ that not all are zero and the map to the projective space is given by the class of $(s_1,cdots, s_{n+1})$, where two such tuples are identified if one is a non-zero multiple of the other one.

This seems to identify $Mor(text{Spec}(F), mathbb{P}^{n})$ with $text{Spec}(F)times mathbb{P}^n$. Then what does $(coprod_n Mor(text{Spec}(F), mathbb{P}^{n}))^+$ look like? How one can recover the expected results from this? Is this supposed to be $Sym^{infty}(text{F}times mathbb{P}^1)$? If so then it implies that the first homotopy group is $H^1(text{Spec}(F),mathbb{Z})$ which is the correct answer but the second homotopy group becomes $mathbb{Z}oplus H^2(text{Spec}(F),mathbb{Z})$ which is incorrect.

Note that the singular cohomology of field is defined as the colimit of singular cohomology of Zariski opens of the variety whose function field is $F$.