Let's $X$ be a projective smooth variety over a field that has an embedding into $\mathbb{C}$. Let's denote the infinite symmetric power of $X$ by $\text{Sym}^{\infty}(X)$. Denote the algebraic simplices by $\Delta^{\bullet}$ and define $C_n(X):=\text{Hom}_{gp}(\Delta^n, \text{Sym}^{\infty}(X))$. Here $gp$ refers to the fact this is group completed i.e. elements of $C_n(X)$ can be written as pairs $(a,b)$ where each $a$ and $b$ are just elements of $\text{Hom}(\Delta^n, \text{Sym}^{\infty}(X))$ and we think of $b$ with a negative sign, the group operation is simply given by monoid structure on the infinite symmetric power. The homology of $C_n(X)$ is called Suslin homology (it is usually stated with language of correspondences). With our assumptions on $X$, assuming $\text{dim}(X)=d$ the $i$-th homology is isomorphic to the following motivic cohomology group $H_{\mathcal{M}}^{2d-i}(X, \mathbb{Z}(d))$.
Now if we focus on the complex points of $X$ with the analytical topology and replace the algebraic simplices with the topological simplices and go through the exact same construction we get the complex $C_n^{top}(X(\mathbb{C}))$ such that its $i$-th homology by Dold-Thom is $H_i(X(\mathbb{C}), \mathbb{Z})$.
What happens if we work with the algebraic simplices and consider the "analytical maps"? An analytic map between $Y$ and $Z$ is defined as giving an analytical cycle on $Y\times Z\times \text{Spec}(\mathbb{C})$ such that surjects onto $Y\times \text{Spec}(\mathbb{C})$ and induces homeomorphism on the complex points with the analytical topology.
