nLab
canonical resolution

Contents

Contents

Idea

Let KK be the strict 2-category Cat and CC a category in KK. Then we can identify a monad on CC with an endofunctor T:CCT:C\to C which is a monoid in K(C,C)K(C,C). We recall that for any object cc of CC, we obtain a certain sort of resolution of cc. Note that the canonical resolution is not a resolution in the sense of the above cited nLab page because it is not necessarily acyclic or contractible in the relevant sense. In other words, it does not always have trivial cohomology, especially since our category may not even be equipped with such a notion.

Definitions

The canonical resolution of cc in CC with respect to a monad T:CCT:C\to C is a coaugmented cosimplicial object CanRes T:Δ +CCanRes_T:\Delta_+\to C such that CanRes T([1])=cCanRes_T([-1])=c and CanRes T([n])=T n+1cCanRes_T([n])=T^{n+1}c for each n0n\geq 0. It has coface? morphisms T ncT n+1cT^{n}c\to T^{n+1}c given by the unit map of TT, and codegeneracies given by the monoid structure TTcTcT T c\to T c.

Monadic Cohomology

Given a suitable functor E:CAE:C\to A, for AA an additive category, we can define the monadic cohomology of an object cc to be the cohomology of the cochain complex associated to the composition of functors ECanRes T:Δ +A.E\circ CanRes_T:\Delta_+\to A. This has been introduced by Godement who called monad and the resolution from it the standard construction (thus, standard resolution is a term which is still used). By suitably dualizing, we can define comonadic homology.

Resolution of a TT-algebra

Suppose that cc is an object of CC and also an algebra for TT. Thus there is a morphism TccT c\to c satisfying certain properties. In particular, this map provides an extra codegeneracy which defines an augmented simplicial object living inside of the canonical resolution of cc. This simplicial object is frequently referred to as the bar construction of cc.

Note that we can consider this internal bar construction to also be the canonical resolution associated to the induced comonad on the Eilenberg-Moore category of algebras over TT.

This bar construction can also be obtained by precomposing CanRes TCanRes_T with the functor that includes Δ + opΔ\Delta^{op}_+\hookrightarrow \Delta as the subcategory whose morphisms preserve the minimal and maximal elements of finite totally ordered sets.

Examples

The bar resolution for abelian groups

There is a monad on Set, the category of sets, comprising the free abelian group functor F:SetAbF \colon Set\to Ab and the forgetful functor U:AbSetU \colon Ab\to Set. The category of algebras of this monad is precisely the category of abelian groups (in other words, the free abelian group functor FF is monadic).

Thus we may produce the cosimplicial canonical resolution of any set XX. If XX supports (at least one) abelian group structure, then we can add a codegeneracy to the canonical resolution which defines the usual bar construction on XX with that particular abelian group structure.

References

Last revised on December 17, 2019 at 10:47:19. See the history of this page for a list of all contributions to it.