Special and general types
In the context of homological algebra, for a chain complex, its chain homology group in degree is akin to the -th homotopy groups of a topological space. It is defined to be the quotient of the -cycles by the -boundaries in .
Dually, for a cochain complex, its cochain cohomology group in degree is the quotient of the -cocycles by the -coboundaries.
Basic examples are the singular homology and singular cohomology of a topological space, which are the (co)chain (co)homology of the singular complex.
Chain homology and cochain cohomology constitute the basic invariants of (co)chain complexes. A quasi-isomorphism is a chain map between chain complexes that induces isomorphisms on all chain homology groups, akin to a weak homotopy equivalence. A category of chain complexes equipped with quasi-isomorphisms as weak equivalences is a presentation for the stable (infinity,1)-category of chain complexes.
Let be an abelian category such as that of -modules over a commutative ring . For the integers this is the category Ab of abelian groups. For a field, this is the category Vect of vector spaces over .
Write for the category of chain complexes in . Write for the category of cochain complexes in .
We label differentials in a chain complex as follows:
For a chain complex and , the chain homology of in degree is the abelian group
given by the quotient (cokernel) of the group of -cycles by that of -boundaries in .
For all forming chain homology extends to a functor from the category of chain complexes in to itself
Chain homology commutes with direct product of chain complexes:
Similarly for direct sum.
Respect for direct sums and filtered colimits
The chain homology functor preserves direct sums:
for and , the canonical morphism
is an isomorphism.
The chain homology functor preserves filtered colimits:
for a filtered diagram and , the canonical morphism
is an isomorphism.
This is spelled out for instance as (Hopkins-Mathew , prop. 23.1).
In the context of homotopy theory
We discuss here the notion of (co)homology of a chain complex from a more abstract point of view of homotopy theory, using the nPOV on cohomology as discussed there.
A chain complex in non-negative degree is, under the Dold-Kan correspondence a homological algebra model for a particularly nice topological space or ∞-groupoid: namely one with an abelian group structure on it, a simplicial abelian group. Accordingly, an unbounded (arbitrary) chain complex is a model for a spectrum with abelian group structure.
One consequence of this embedding
induced by the Dold-Kan nerve is that it allows to think of chain complexes as objects in the (∞,1)-topos ∞Grpd or equivalently Top. Every (∞,1)-topos comes with a notion of homotopy and cohomology and so such abstract notions get induced on chain complexes.
Of course there is an independent, age-old definition of homology of chain complexes and, by dualization, of cohomology of cochain complexes.
This entry describes how these standard definition of chain homology and cohomology follow from the general (∞,1)-topos nonsense described at cohomology and homotopy.
The main statement is that
Before discussing chain homology and cohomology, we fix some terms and notation.
In a given (∞,1)-topos there is a notion of homotopy and cohomology for every (co-)coefficient object ().
The particular case of chain complex homology is only the special case induced from coefficients given by the corresponding Eilenberg-MacLane objects.
Assume for simplicity here and in the following that we are talking about non-negatively graded chain complexes of vector spaces over some field . Then for every write
for the th Eilenberg-MacLane object.
Notice that this is often also denoted or or .
Homotopy and cohomology
With the Dold-Kan correspondence understood, embedding chain complexes into ∞-groupoids, for any chain complexes , and we obtain
whose * objects are the -valued cocycles on ; * morphisms are the coboundaries between these cocycles; * 2-morphisms are the coboundaries between coboundaries * etc. and where the elements of are the cohomology classes
whose * objects are the -co-valued cycles on ; * morphisms are the boundaries between these cycles; * 2-morphisms are the boundaries between boundaries * etc. and where the elements of are the homotopy classes
Chain homology as homotopy
For any chain complex and its ordinary chain homology in degree , we have
A cycle is a chain map
Such chain maps are clearly in bijection with those elements that are in the kernel of in that .
A boundary is a chain homotopy
such that .
Cohomology of cochain complexes
The ordinary notion of cohomology of a cochain complex is the special case of cohomology in the stable (∞,1)- category of chain complexes.
For a cochain complex let
be the corresponding dual chain complex. Let
be the chain complex with the tensor unit (the ground field, say) in degree and trivial elsewhere. Then
as objects chain morphisms
These are in canonical bijection with the elements in the kernel of of the dual cochain complex .
as morphism chain homotopies
Comparing with the general definition of cocycles and coboudnaries from above, one confirms that
the cocycles are the chain maps
the coboundaries are the chain homotopies between these chain maps.
the coboundaries of coboundaries are the second order chain homotopies between these chain homotopies.
Basics are for instance in section 1.1 of