Contents
Context
Category theory
Higher algebra
Contents
Idea
One usually defines cohomology with respect to some coefficient objects:
Beck modules are a simultaneous generalisation of all three types of module.
Definition
Let be a category with pullbacks and let be an object in . A Beck module over is an abelian group object in the slice category . In particular, if is the terminal object, this reduces to the notion of an abelian group object in . We write for the category of Beck modules over .
(Beck 67, def. 5)
Properties
Proof
If is a effective regular category (resp. locally presentable category), then so is . Thus, the claim reduces to the fact that the category of abelian group objects in an effective regular category (resp. locally presentable category) is an abelian category (resp. locally presentable category).
Proposition
Let be an effective regular category with filtered colimits and let be an object in . If filtered colimits in preserve finite limits, then (is an abelian category and) satisfies axiom AB5.
Proof
The forgetful functor creates pullbacks and filtered colimits, so filtered colimits in also preserve finite limits. The forgetful functor creates limits and filtered colimits, so filtered colimits in preserve kernels. In view of the earlier proposition, it follows that satisfies axiom AB5.
Corollary
Let be a locally finitely presentable effective regular category and let be an object in . Then is a Grothendieck category.
Proof
Combine the two propositions above.
Derivations
Let be a category with pullbacks, let be an object in , and let be the forgetful functor. Given a Beck module over , an -valued derivation of is a morphism in , where is the terminal object in , and we write
for the set of -valued derivations of . The Beck module of differentials over is an object in representing the functor .
The Beck module is not guaranteed to exist in general. When the functor has a left adjoint, is simply the value of the left adjoint at .
Proposition
Let be a locally presentable category and let be an object in . Then the forgetful functor has a left adjoint.
Proof
The forgetful functor creates limits and -filtered colimits (for some large enough), so we may apply the accessible adjoint functor theorem.
Examples
Beck modules over associative algebras
Proposition
Let be the category of (not necessarily commutative) rings and let be a ring. Then is equivalent to the category of -bimodules.
Proof
Let be ring homomorphism. To give it the structure of a Beck module over , we must give ring homomorphisms and such that , , as well as various other equations. Given elements of such that and , we have the following interchange law:
Hence, if ,
but , so
and we conclude that
and in particular, is entirely determined by and . We also have the following interchange law,
and in particular,
hence,
so if , then .
Let . The above shows that the internal abelian group structure on restricts to the pre-existing abelian group structure on . In addition, the homomorphism gives the structure of an -bimodule, and we see that is naturally isomorphic to the square-0 extension ring , with componentwise addition and the multiplication given below,
regarded as a Beck module over by defining , , and as follows:
Thus, we have an equivalence between and the category of -bimodules, as claimed.
Proposition
Let be a ring. Then the Beck module is isomorphic to the -bimodule of Kähler differentials (relative to ).
Proof
Let be an -bimodule, regard as a ring as above, and let be the obvious projection. A ring homomorphism satisfying is the same thing as an additive homomorphism satisfying the following equations,
i.e. a derivation (over ). Thus, the Beck module has the same universal property as the -bimodule of Kähler differentials.
Beck modules over groups
Proposition
Let be the category of (not necessarily abelian) groups and let be a group. Then is equivalent to the category of left -modules.
Proof
Let be group homomorphism. To give it the structure of a Beck module over , we must give group homomorphisms and such that , , as well as various other equations. Given elements of such that and , we have the following interchange law:
and in particular,
but on the other hand, if , then
and writing for the unit of and , we have , hence
so we conclude that
and in particular, is entirely determined by .
Let . The above shows that the internal abelian group structure on restricts to the pre-existing group structure on . (In particular, is an abelian group!) We make into a left -module as follows:
We can then construct the semi-direct product , which has the following multplication:
There is a group homomorphism defined by , and it is bijective: surjectivity is clear, and injectivity is a consequence of the fact that . We may regard as a Beck module over by defining , , and as follows:
Thus, we have an equivalence between and the category of left -modules, as claimed.
Proposition
Let be a group and let be a left -module. Under the above identification of Beck modules over with left -modules, -valued derivations of are precisely crossed homomorphisms , i.e. maps satisfying the following equation:
Proof
Let be the evident projection. A group homomorphism such that is the same thing as a map satisfying the equation below,
which is equivalent to the defining equation for crossed homomorphisms.
The tangent category
One may assemble the individual categories of Beck modules over the objects of into a category fibred over , called the tangent category.
References
The concept is due to
- Jon Beck, Triples, algebras and cohomology, Ph.D. thesis, Columbia University, 1967, Reprints in Theory and Applications of Categories, No. 2 (2003) pp 1-59 (TAC)
and was popularized in
- Daniel G. Quillen, On the (co-)homology of commutative rings, in Proc. Symp. on Categorical Algebra, 65 – 87, American Math. Soc., 1970.
See also
An application to knot theory is given in
- Markus Szymik, Alexander-Beck modules detect the unknot, Fund. Math. 246 (2019) 89-108.