homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(also nonabelian homological algebra)
The standard interval object in a category of chain complexes in Mod is an “abelianization” of the standard simplicial interval, the 1-simplex and a model of the unit interval, , with the evident cell decomposition.
Let be some ring and let Mod be the abelian category of -modules. Write for the corresponding category of chain complexes.
The standard interval object in chain complexes
is the normalized chain complex of the simplicial chains on the simplicial 1-simplex:
In components this means that
A homotopy with respect to gives a chain homotopy and conversely.
See the entry on chain homotopy for more details.
Last revised on September 3, 2012 at 13:21:36. See the history of this page for a list of all contributions to it.