TITLE: Are Anderson $T$-motives motives for the function field analogy?
QUESTION [5 upvotes]: this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question. 
Let $\mathbb{C}_{\infty}$ be the function field analog of $\mathbb{C}$ for $\mathbb{F}_q (\theta) = \mathbb{Q}_{\infty}$ and $A = \mathbb{C}_{\infty} [T, \tau]$ the Anderson ring ( i.e.,  $T$ is central and $\tau$ acts as the Frobenius endomorphism).
An Anderson $T$-motive is simply a left $A$-module $M$ which is free and finitely generated over $\mathbb{C}_{\infty} [\tau]$, and satisfies $$(T - \theta)^n M/\tau M = \{ 0 \}$$ for some  $n > 0$. If, furthermore, $M$ is finitely generated over $\mathbb{C}_{\infty}[T]$ (or equivalently, free of finite rank), then it's called an abelian $T$-motive or $t$-motive.
First of all, why the word "motives" in "Anderson $T$-motives"?
Are $T$-motives analogous to motives of arithmetic schemes? What's its number field analogue?
Thanks in advance.

REPLY [6 votes]: This is just a small supplement to Wendt's (as usual) excellent response.
Drinfel'd modules (or elliptic modules as Drinfel'd called them) are the function field analogues of elliptic curves. Note that elliptic curves make sense also over function fields. So elliptic modules are exotic analogues of elliptic curves. Similarly, Anderson t-motives are the function field analogues of the Tannakian category of (mixed) motives generated by abelian varieties. The Tannakian word is because the category of t-motives has a tensor product and duals, abelian varieties because the category contains elliptic modules.
But even this is not quite accurate. Because only Drinfeld modules of rank two correspond to elliptic curves, and those of rank one correspond to Tate motive (or its twists), and (it is tempting to think) those of rank >2 as corresponding to non-commutative tori (these are not connected to motives, as far as I know).
So Anderson t-motives is the function field analogue of the Tannakian category of mixed motives generated by abelian varieties, Artin-Tate motives, and ...
Here motives would mean under homological/numerical equivalence. There is no underlying geometry for Anderson's t-motives (varieties whose Chow groups etc could give t-motives).
@Wendt: it is only over finite fields that abelian varieties generate the category of motives. this does not hold over number fields.