nLab
model structure on dg-comodules
Contents
Context
Model category theory
model category

Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$ -categories
Model structures
for $\infty$ -groupoids
for ∞-groupoids

for equivariant $\infty$ -groupoids
for rational $\infty$ -groupoids
for rational equivariant $\infty$ -groupoids
for $n$ -groupoids
for $\infty$ -groups
for $\infty$ -algebras
general
specific
for stable/spectrum objects
for $(\infty,1)$ -categories
for stable $(\infty,1)$ -categories
for $(\infty,1)$ -operads
for $(n,r)$ -categories
for $(\infty,1)$ -sheaves / $\infty$ -stacks
Contents
Idea
A model category structure on the category of comodules in chain complexes over a dg-coalgebra .

Details
Let $C$ be a differential graded-cocommutative coalgebra over a field .

Model structure of the second kind

There exists a model category structure on the category $C dgCoMod$ of dg-comodules over $C$ whose

This is due to (Positelski 11, 8.2 Theorem (a) ).

Model structure of the first kind

There is another model structure where the fibrations in addition satisfy the condition that their kernel $K$ satisfies that for all acyclic $N$ , then $\underline{Hom}_C(N,K)$ is acyclic.

This is due to (Positelski 11, 8.2 Remark ), there called the “model category structure of the first kind”. This is also reviewed as (Pridham 13, prop. 2.2 ).

References
Last revised on June 6, 2017 at 12:22:12.
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