nLab simplicial resolution

Idea

The concept of resolution in homotopy theory specialized to simplicial homotopy theory is “simplicial resolution”. Simplicial resolutions can be constructed in various ways, for instance, by a comonad, or by a step-by-step method, developed by Michel André, that resembles the construction of a Eilenberg-Mac Lane space from a group presentation, followed by adding cells to ‘kill’ the higher homotopy groups.

the following needs attention

Remark

The term simplicial resolution is also used more generally.

In any ambient category or $\infty$-category $C$ that admits a notion of colimit or weak colimit, a simplicial resolution of an object $c \in C$ is a simplicial object $y_\bullet : \Delta^{op} \to C$ together with a realization of $c$ as a colimit

$$c \simeq colim_{[k]} y_k \,.$$

The term is also used for a Reedy fibrant replacement of a constant simplicial object in a model category; see also resolution.

Cocones under a simplicial diagram are exactly the augmented simplicial objects, so a simplicial resolution of $c$ can alternately be described as an augmented simplicial object $y_\bullet : \Delta^{op}_+ \to C$ that is a colimiting cocone, and such that $y_{-1} = c$.

Examples

Free simplicial resolution of a group.

There is the obvious adjoint pair of functors,

$$U:Groups\to Sets, F:Sets\to Groups.$$

Writing $\eta : Id \to UF$ and $\varepsilon :FU\to Id$ for the unit and counit of this adjunction, we have a comonad on $Groups$, the free group comonad, $( FU, \varepsilon, F\eta U)$.

We write $L = FU$, $\delta = F\eta U$, so that

$$\varepsilon : L \to I$$

is the counit of the comonad whilst

$$\delta: L \to L^2$$

is the comultiplication.

Now suppose $G$ is a group and set $F(G)_i = L^{i+1}(G)$, so that $F(G)_0$ is the free group on the underlying set of $G$ and so on.

The counit (which is just the augmentation morphism from $FU(G)$ to $G$) gives, in each dimension, face morphisms

$$d_i = L^{n-i}\varepsilon L^{i}(G) : L^{n+1}(G) \to L^n(G),$$

for $i = 0, \ldots, n$, whilst the comultiplication gives degeneracies,

$$s_i : L^n(G) \to L^{n+1}(G)$$

$$s_i = L^{n-1-i}\delta L^{i},$$

for $i = 0, \ldots, n-1$,

satisfying the simplicial identities.

The resulting object is an augmented simplicial group that is free in all non-negative dimensions and is acyclic. It has zeroth homotopy equal to $G$ and all homotopy groups are trivial.

Of course this construction did not depend on the fact that we were handling groups, so we could apply it to any comonad on any category (within reason!) This has the advantage of providing simplicial resolutions that are functorial. These are sometimes called comonadic resolutions. Working with monads on a category gives, a cosimplicial object which is a cosimplicial resolution given any object.

This leads to the subject of monadic cohomology.

A cautionary note is in order. The simplicial resolution of an object derived from a comonad is sometimes presented in an opposite form, so

$$d_i = L^i\varepsilon L^{n-i}(G) : L^{n+1}(G) \to L^n(G),$$

for $i = 0, \ldots, n$, and similarly for the degeneracies. This is more or less equivalent to the form given here as it just uses the opposite simplicial object.

Cech nerve

In an (infinity,1)-topos Čech cover $C(U) \stackrel{\simeq}{\to} X$ induced by a cover $(U = \coprod_i U_i) \to X$ is a simplicial resolution of $X$.

Related concepts

• homological resolution

• monadic cohomology

• canonical resolution

References and Literature

Classical soureces for the comonadic form include Godement‘s book,

• Roger Godement, Topologie Algébrique et Théorie des Faisceaux. Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13 Hermann, Paris 1958.

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,

and in Jack Duskin’s monograph:

• J. Duskin, 1975, Simplicial methods and the interpretation of “triple” cohomology , number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc.

The step-by-step method of producing a simplicial resolution was developed by Michel André in the texts:

• Michel André, Méthode simpliciale en algèbre homologique et algèbre commutative, Springer Lecture Notes in Mathematics, Vol 32, 1967.

• Michel André, Homologie des algèbres commutatives Grundlehren der mathematischen Wissenschaften, Band 206. Springer. 1974

with a view to applications in commutative algebra and in particular in the development of the cohomology known as André-Quillen cohomology and the study of the cotangent complex.

Simplicial resolutions in the context of presentable (infinity,1)-categories are discussed in section 6.1.4 of

• Jacob Lurie, Higher Topos Theory

(below lemma 6.1.4.3)