nLab canonical resolution

Idea

Let $K$ be the strict 2-category Cat and $C$ a category in $K$. Then we can identify a monad on $C$ with an endofunctor $T \colon C\to C$ (a monoid in the endofunctor category $K(C,C)$). We recall that for any object $c$ of $C$, we obtain a certain sort of resolution of $c$. 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 $c$ in $C$ with respect to a monad $T \colon C\to C$ is a coaugmented cosimplicial object $CanRes_T:\Delta_+\to C$ such that $CanRes_T([-1])=c$ and $CanRes_T([n])=T^{n+1}c$ for each $n\geq 0$. It has coface? morphisms $T^{n}c\to T^{n+1}c$ given by the unit map of $T$, and codegeneracies given by the monoid structure $T T c\to T c$.

Monadic Cohomology

Given a suitable functor $E:C\to A$, for $A$ an additive category, we can define the monadic cohomology of an object $c$ to be the cohomology of the cochain complex associated to the composition of functors $E\circ CanRes_T:\Delta_+\to A.$ 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 $T$-algebra

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

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 $T$.

This bar construction can also be obtained by precomposing $CanRes_T$ with the functor that includes $\Delta^{op}_+\hookrightarrow \Delta$ 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 $F \colon Set\to Ab$ and the forgetful functor $U \colon Ab\to Set$. The category of algebras of this monad is precisely the category of abelian groups (in other words, the free abelian group functor $F$ is monadic).

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

Related concepts

• homological resolution

• monadic cohomology

• simplicial resolution

• bar and cobar construction

References

The original articles:

• Roger Godement, Appendix of: Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

• Peter J. Huber, Homotopy theory in general categories, Mathematische Annalen 144 (1961) 361–385 [doi:10.1007/BF01396534]

The theory is explained in many sources, including

• M. Barr and J. Beck, 1969, Homology and Standard Constructions, in Seminar on triples and categorical homology, number 80 in Lecture Notes in Maths., Springer-Verlag, Berlin,

• John Duskin, Simplicial methods and the interpretation of “triple” cohomology“ Memoirs of the AMS 163, Amer. Math. Soc. (1975) [ISBN:978-1-4704-0645-5]

Textbook account:

• Charles Weibel, An Introduction to Homological Algebra