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What is a Non-example of class structure??

User Myol
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2 Answers

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Answer:

the oppsite

Step-by-step explanation:

class structure. The hierarchical organization by which a society or community is divided into classes. The vast majority of the population of the United States considers itself as belonging to the middle class.

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Let CDGA be the category of commutative differential graded algebras over a fixed ground field k of characteristic
p
. Weak equivalences are quasi-isomorphisms, fibrations are levelwise surjections. These would determine the others, but cofibrations are essentially generated by maps →
A

B
such that on the level of the underlying DGA,
B
is a polynomial algebra over
A
on a generator
x
whose boundary is in
A
.

CDGA is complete and cocomplete, satisfies the 2
2
-out-of-3
3
axiom, the retract axiom, satisfies lifting, and a general map can be factored into a cofibration followed by an acyclic fibration by the small object argument.

However, you don't have factorizations into acyclic cofibrations followed by fibrations, because of the following.

Suppose →
A

B
is a map of commutative DGAs which is a fibration in the above sense. Then for any element []
[
x
]
in the (co)homology of
B
in even degree, the
p
-th power []
[
x
]
p
is in the image of the cohomology of
A
. In fact, pick any representing cycle ∈
x

B
and choose a lift ∈
y

A
. Then the boundary of
y
p
is −1=0
p
y
p

1
=
0
by the Leibniz rule, so []
[
y
p
]
is a lift of []
[
x
]
p
to the (co)homology of
A
.

(As a result, there are a lot of other "homotopical" constructions, such as homotopy pullbacks, that are forced to throw you out of the category of commutative DGAs into the category of ∞
E

DGAs.)

Nothing goes wrong in characteristic zero.
User Vlad Skurtolov
by
5.4k points