Question

1. Short note on:
a. Duality of motive​

Answer 1

Explanation:

This theory posits that two competing psychological forces shape party identification—partisan motivation and responsiveness motivation. On one hand, partisans are driven to maintain party loyalty, but on the other hand, they are motivated to be responsive to their political environment. Because each voter only has a single vote to cast, voting behavior has essentially zero impact on the policy benefits the voter receives. This means that there are essentially no policy benefits to be gained from adjusting one’s party identity to reflect one’s disagreements. Therefore, responsiveness motivation is thought to be primarily driven the by the desire to see oneself as a good citizen and not merely a biased partisan. To reconcile their competing motives, partisans must find ways to justify their party identity when disagreements arise. Party identification change occurs when a justification cannot be manufactured or when responsiveness motivation is sufficient to outweigh partisan motivation

Answer 2

Duality of Motive-

Recently, due to the active study of cohomological invariants in algebraic geometry,

“transplantation” of classical topological constructions to the algebraic-geometrical “soil”

seems to be rather important. In particular, it is very interesting to study topological

properties of the category of motives.

The concept of a motive was introduced by Alexander Grothendieck in 1964 in order

to formalize the notion of universal (co-)homology theory (see the detailed exposition of

Grothendieck’s ideas in [5]). For us, the principal example of this type is the category of

motives DM−, constructed by Voevodsky [12] for algebraic varieties.

The Poincar´e duality is a classical and fundamental result in algebraic topology that

initially appeared in Poincar´e’s first topological memoir “Analysis Situs” [9] (as a part of

the Betti numbers symmetry theorem proof). The proof of the general duality theorem

for extraordinary cohomology theories apparently belongs to Adams [1].

Our purpose in this paper is to establish a general duality theorem for the category of

motives. Essentially, we extend the main statement of [8] to this category. Many known

results can easily be interpreted in these terms. In particular, we get a generalization of

the Friedlander–Voevodsky duality theorem [4] to the case of the ground field of arbitrary

characteristic. The proof of this fact, involving the main result of [8], was kindly conveyed

to the authors by Andreˇı Suslin in a private communication.

Being inspired by his work and Dold–Puppe’s category approach [2] to the duality

phenomenon in topology, we decided to present a short, simple, and self-contained proof

of a similar result for the category of motives.

Our result might be viewed as a purely abstract theorem and rewritten in the spirit

of “abstract nonsense” as a statement about some category with a distinguished class

of morphisms. Essentially, what is required for the proof is the existence of finite fiber

products and the terminal object in the category of varieties, a small part of the tensor

triangulated category structure for motives, and finally, the existence of transfers for the

class of morphisms generated by graphs of a special type (of projective morphisms).