Nonabelian Algebraic Topology


Higher category theory

higher category theory

Basic concepts

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Universal constructions

Extra properties and structure

1-categorical presentations

This entry is about the book

  • Ronnie Brown, Philip Higgins, Rafael Sivera, Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids Tracts in Mathematics 15, European Mathematical Society , web, from which the full pdf is available.

The publication details of the book are as follows:

ISBN 978-3-03719-083-8, DOI 10.4171/083

August 2011, 703 pages, hardcover, 17 x 24 cm.

98.00 €

(distribution via the AMS in October, 2011)

The EMS allows a full pdf (with hyperref) on Ronnie Brown’s web page, web.

This treats algebraic topology using tools of strict ω-groupoid-theory: notably the traditional homological algebra use of chain complexes of abelian groups is generalized to crossed complexes, and emphasis is put on the notion of fundamental groupoid and its strict higher categorical generalizations to the cubical fundamental omega-groupoid? of a filtered space over the bare homotopy groups of a space.

One of the main motivations for the development of Nonabelian Algebraic Topology was the observation that the Seifert-van Kampen theorem is most naturally understood as being not about homotopy groups, but about the cubical fundamental omega-groupoid? of a filtered space and may be generalized to a higher homotopy van Kampen theorem this way.

The restriction to strict ∞-groupoids/crossed complexes is still a severe restriction as compared to the full homotopy theory of topological spaces but already more general than the strict and strictly abelian \infty-groupoids used in traditional algebraic topology in the guise of chain complexes of abelian groups. In terms of the cosmic cube of higher category theory the approach of Nonabelian algebraic topology used here is somewhere half way between homology and homotopy theory; it in this border area that traditional accounts seem to most lacking, and are unable to cope well with the nonabelian second relative homotopy group of a pair of spaces. The start of the new approach is to replace this by a homotopy double groupoid of a pair of spaces, which allows an algebraic inverse to subdivision.



Some comments from Ronnie Brown himself:

I hope it is helpful to relate my experiences from the 1960s and later with nonabelian cohomology.

In writing my book on topology in the 1960s, I got offended by having to make a detour to get the fundamental group of the circle, and then was attracted by Paul Olum?’s paper referenced below. I extended Olum’s work to a Mayer–Vietoris type sequence in the second paper below, and this enabled one to compute the fundamental group of, for example, a wedge of circles.

(I use an MV sequence in Topology and Groupoids in connection with pullbacks of covering spaces.)

So I decided to use this account for the book, thus giving students the advantage, it seemed, of an introduction to cohomological ideas.

The problem was that the account when written in detail came to 30 pages (or maybe 40) and when looked at in the cold light of day seemed incredibly boring (a full account is different from Olum’s research account).

I was at the time looking for exercises and came across Philip Higgins’ paper on presentations of groupoids, which used free products with amalgamation of groupoids. So I decided to give an exercise on the fundamental groupoid of a union. Then I felt I ought to write out a solution. When I had done this, it seemed streets ahead in exposition of all that nonabelian cohomology stuff and moreover, when souped up to the fundamental groupoid on a set of base points , gave results not reachable by the MV sequence; for example you could not with the MV sequence deduce the precise calculation of the fundamental group of a union of two open sets whose intersection had say 150 path components. (This anomaly is also significant, in illustrating the limitations of exact sequences.)

So I decided to switch to an exposition of groupoids in 1-dimensional homotopy theory (also spurred by a meeting with George Mackey in 1967 where he told me of his work on ergodic groupoids, which is now seen as a preliminary to Noncommutative Geometry).

It occurred to me that if one could come to the groupoid idea from two distinct directions, then there was likely to be more in this than met the eye. At the same time, an examination of the proof of the van Kampen theorem for groupoids, suggested that the theorem should have an extension to all dimensions, if one could define homotopy gadgets with the right properties. Another stimulus was the proof (used in the book) by Frank Adams (circulated in handwritten lecture notes) of the cellular approximation theorem, which had analogies to parts of the van Kampen proof, but failed to get algebraic results because, apparently, of the lack of an appropriate algebraic gadget in dimension n>1n \gt 1.

It took 9 years to find such a gadget in dimension 22, and another 3 to get them in all dimensions, in work with Philip Higgins.

It seemed to me unfortunate that this work aroused the opposition, for reasons never explained to me, of Frank Adams, who told people the whole programme was “ridiculous”. His opinion became the opposite only when I told him (1985?) of the extension to the non simply connected case of the Blakers-Massey description of π 3\pi_3 of a triad, using the nonabelian tensor product (work with Jean-Louis Loday).

The higher order van Kampen theorems, and the often nonabelian calculations which result, have not been obtained by cohomological methods, but only by working directly with structures appropriate to the geometry of higher homotopies, i.e. forms of strict multiple groupoids. This confirms the comment of Philip Hall?, Philip Higgins’ supervisor, that one should not try to force the geometry into a given algebraic mode, but search for the algebra which models the geometry. So it seems to me that algebraic topology has been mainly restricted to, or not got out of, the single base point and “group”, not “groupoid”, mode, nor appreciated the possibilities of colimit type theorems in algebraic (and geometric?) topology – no algebraic or geometric topology text (except mine!) mentions the higher order van Kampen work with Philip Higgins.

You can also see this restriction in the contrast between the unsymmetrical, choice laden, definition of the second relative homotopy group, with its compositions in one direction (recall the limitations of “Lineland” described in “Flatland”) and the definition of the fundamental double groupoid of a pointed pair of spaces ρ 2(X,A)\rho_2(X,A), with its compositions in 22 directions. This contrast gets more significant in higher dimensions.

For all these reasons, my inclination is to look for the applications of the “appropriate” (whatever that is!) structures rather than cohomology with coefficients in such structures, where lots of detail is likely to get lost. Also, in making calculations it is convenient to work with strict algebraic structures, where the notion of colimit is more comprehensible. Even there, it has been a problem to make say colimit calculations with crossed modules into a symbolic computer algebra format. See the work by Chris Wensley? listed below.

These results could not have been obtained without the intuitions on multiple compositions easily allowed by a cubical approach.

One of the key observations for this programme was that one could define a strict homotopy double groupoid for a pointed pair of spaces, and that this was closely related to the well known fundamental crossed module of a pair of spaces, first considered by J.H.C. Whitehead. His paper listed below was a key source of ideas.

The natural extension of this observation is to construct a strict cubical ω-groupoid? ρX *\rho X_* of a filtered space X *X_*, and find its relation to the quite classical homotopically defined fundamental crossed complex functor Π:(filteredspaces)(crossedcomplexes)\Pi: (filtered spaces) \to (crossed complexes). The proofs here are non trivial. By proving using ρ\rho a colimit theorem for Π\Pi one can shortcut singular homology, and obtain old and new results in algebraic topology, including some explicit calculations of homotopy groups, even as modules over the fundamental group. This working with filtered space is not unreasonable since they abound. For example, classifying spaces often come with convenient filtrations, as do geometric realisations of simplicial or cubical sets. These ideas generalise of course to multifiltered space?s or nn-cubes of spaces. It is not so clear that one must work with a kind of bare topological space, and so have little handle on which to construct invariants, except say by first taking a singular complex, or using multipaths.

The main idea of the higher homotopy van Kampen Theorems is to model algebraically the gluing of homotopy types, or limited models of such.

An indication of a beginnings of a Čech type approach to nonabelian cohomology using groupoids and crossed complexes is given in the new book, Chapter 12. This has not been developed in terms of sheaf theory.

Another big gap in comparison with traditional algebraic topology is intersection theory and Poincare duality, although the (quite complicated) machinery of tensor products is available in the crossed complex context.

An obvious gap is also that of extending Grothendieck’s work on the fundamental group!


I 1 and 2-dimensional results

Crossed complexes

7 The basics of crossed complexes

7.1 Our basic categories and functors
7.1.i The category of filtered topological spaces
7.1.ii Modules over groupoids

A module over a groupoid is a collection of abelian groups equipped with a linear action by a groupoid.


(module over a groupoid)

Let 𝒢=(𝒢 1𝒢 0)\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0) be a groupoid. A module over the groupoid 𝒢\mathcal{G} is a collection {N x} x𝒢 0\{N_x\}_{x \in \mathcal{G}_0} of abelian groups equipped with a collection of maps

N x×𝒢(x,y)N y N_x \times \mathcal{G}(x,y) \to N_y

that are linear and respect the groupoid composition in the obvious way.

7.1.iii The category of crossed complexes
7.1.iv Homotopy and homology groups of crossed complexes


7.1.v The fundamental crossed complex functor

The notion of fundamental groupoid of a topological space generalizes to a notion of fundamental ∞-groupoid. There is a strict version of this (which loses some information): the fundamental strict \infty-groupoid. In a filtered space X *X_* one can consider the variant where the k-morphisms of the fundamental \infty-groupoid are constrained to lie in X kX_k. The fundamental crossed complex of a filtered space is the equivalent crossed complex incarnation of the fundamental strict \infty-groupoid of a filtered space.


(fundamental crossed complex)

Let X X_\bullet be a filtered space.

Write Π 1(X 1,X 0)\Pi_1(X_1,X_0) for the subgroupoid of the fundamental groupoid Π 1(X 1)\Pi_1(X_1) of X 1X_1 on objects that are in X 0X_0.

The fundamental crossed complex ΠX *\Pi X_* of XX is the crossed complex


(ΠX *) 1=Π 1(X 1,X 0) (\Pi X_*)_1 = \Pi_1(X_1,X_0)


(ΠX *) n:= xX 0π n(X n,X n1,x)forn2, (\Pi X_*)_n := \coprod_{x \in X_0} \pi_n(X_n, X_{n-1}, x) \;\;\;\; for n \geq 2 \,,

where π n(X n,X n1,x)\pi_n(X_n, X_{n-1}, x) is the relative homotopy group? obtained by equivalence classes of maps from the pointed nn-disk into XX such that the disk lands in X nX_n, its boundary in X n1X_{n-1} and its basepoint on xx.

See also section 1 of


(fundamental crossed complex of the nn-simplex)

The topological nn-simplex Δ n\Delta^n is canonically a filtered space with (Δ n) k(\Delta^n)_k being the union of its kk-faces.

Then we have that Π 1((Δ n) 1,(Δ n) 0)\Pi_1((\Delta^n)_1, (\Delta^n)_0) is the groupoid whose objects are the n+1n+1 vertices of Δ n\Delta^n and which has precisely one morphism x ix jx_i \to x_j for each ordered pair x i,x j(Δ n) 0x_i,x_j \in (\Delta^n)_0 (all of them being isomorphisms)

Π 1((Δ 2) 1,(Δ 2) 0)={ x 1 x 0 x 2}. \Pi_1((\Delta^2)_1,(\Delta^2)_0) = \left\{ \array{ && x_1 \\ & \nearrow\swarrow && \searrow \nwarrow \\ x_0 &&\stackrel{\leftarrow}{\to}&& x_2 } \right\} \,.

At any x ix_i the relative homotopy group π 2((Δ n) 2,(Δ n) 1,x i)\pi_2((\Delta^n)_2,(\Delta^n)_1, x_i) is a group on the set of 2-faces that have x ix_i as a 0-face: there is a unique homotopy class of disks in Δ n\Delta^n that sits in the 2-faces (Δ n) 2(\Delta^n)_2, whose base point is at x jx_j and whose boundary runs along the boundary of a given 2-face of Δ n\Delta^n.

So (using the equivalence of crossed complexes with strict ω\omega-groupoids) for instance ΠΔ 2\Pi \Delta^2 is generated from Π 1((Δ 2) 1,(Δ 2) 0)\Pi_1((\Delta^2)_1,(\Delta^2)_0) as above and a 2-cell

x 1 x 0 x 2 \array{ && x_1 \\ & \swarrow &\Downarrow& \nwarrow \\ x_0 &&\to&& x_2 }

under whiskering and composition. For instance whiskering this with x 1x 2x_1 \to x_2 yields the 2-morphism

x 1 x 0 x 2. \array{ && x_1 \\ & \swarrow &\swArrow& \searrow \\ x_0 &&\to&& x_2 } \,.

One sees that ΠΔ 2\Pi \Delta^2 is the strict groupoidification of the second oriental.

Generally, ΠΔ n\Pi \Delta^n is the nn-groupoid freely generated from kk-morphisms for each kk-face of Δ n\Delta^n.

7.4 Crossed complexes and chain complexes

(groupoid module chain complexes)

Write ChnChn for the category of chain complexes of modules over a groupoid.

This is Def. 7.4.1.


(groupoid module chain complexes)

Write CrsCrs for the category of crossed complexes.

7.4.i Adjoint module and augmentation module

Given a module over a groupoid (N,𝒢)(N,\mathcal{G}), the semidirect product groupoid 𝒢N\mathcal{G} \ltimes N has the same objects as 𝒢\mathcal{G} and morphisms

(𝒢N)(p,q)=𝒢(p,q)N(q) (\mathcal{G} \ltimes N)(p,q) = \mathcal{G}(p,q) \ltimes N(q)

with composition given by the action of 𝒢\mathcal{G} on NN.

This is def. 7.4.5


(covering morphism)

For (t:N𝒢 0,𝒢)(t : N \to \mathcal{G}_0,\mathcal{G}) a module over a groupoid, write P(N,𝒢)P(N,\mathcal{G}) for the groupoid 𝒢\mathcal{G} pulled back to the underlying set of NN:

an object of P(N,𝒢)P(N,\mathcal{G}) is an element in NN and a morphism n 1n 2n_1 \to n_2 is a morphism 𝒢(t(n 1),t(n 2))\mathcal{G}(t(n_1),t(n_2)).

This is def. 7.4.9.

7.4.iii The derived chain complex of a crossed complex

(chain complex from a crossed complex)

Define a functor :CrsChn\nabla : Crs \to Chn from crossed complexes to modules over groupoids as follows:

For CC a crossed complex we set for n3n \geq 3

(C) n:=C nforn3 (\nabla C)_n := C_n \;\;\;\; for n \geq 3

and for n2n \leq 2 it is given by

This is definition 7.4.20.

7.4.v The right adjoint of the derived functor

We describe a construction of a crossed complex from a chain complex of modules over a groupoid (A n,)(A_n, \mathcal{H}). As a special case it in particular gives an map of ordinary chain complexes of abelian groups into the category of crossed complexes, and hence into strict ω-groupoids.

Recall the definition of the semidirect product groupoid A n\mathcal{H} \ltimes A_n.


(crossed complex from a chain complex)

For AA a chain complex of modules over a groupoid \mathcal{H}, let ΘACrs\Theta A \in Crs be the crossed complex

ΘA:=κ *ΘA, \Theta A := \kappa^* \Theta' A \,,


ΘA:=[A n nA n1A 3 3A 2(0, 2)A 1] \Theta' A := \left[ A_n \stackrel{\partial_n}{\to} A_{n-1} \stackrel{}{\to} \cdots \stackrel{}{\to} A_{3} \stackrel{\partial_3}{\to} A_2 \stackrel{(0,\partial_2)}{\to} \mathcal{H}\ltimes A_1 \right]

and where

κ:P(A 0,)A 0 \kappa : P(A_0, \mathcal{H}) \to \mathcal{H} \ltimes A_0

is the canonical covering morphism from above.

(ΘA) 3 (ΘA) 2 (ΘA) 1 P(A 0,) A 3 3 A 2 (0, 2) A 1 (1, 1) A 0. \array{ \cdots \to & (\Theta A)_3 &\to& (\Theta A)_2 &\to& (\Theta A)_1 &\to& P(A_0, \mathcal{H}) \\ & \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots \to & A_3 &\stackrel{\partial_3}{\to}& A_2 &\stackrel{(0,\partial_2)}{\to}& \mathcal{H} \ltimes A_1 &\stackrel{(1, \partial_1)}{\to}& \mathcal{H} \ltimes A_0 } \,.

Here A 1\mathcal{H} \ltimes A_1 acts on A nA_n for n2n \geq 2 via the projection A 1\mathcal{H} \ltimes A_1 \to \mathcal{H}, i.e. A 1A_1 acts trivially. (…)

Finally set Θ(A) 0:=A 0\Theta(A)_0 := A_0.

We spell out what this boils down to explicitly.

Explicit description

Let A A_\bullet be a chain complex of modules over the groupoid \mathcal{H}. Then the crossed complex Θ(A)\Theta(A) is the following.

  • Its set of objects is Θ(A) 0=A 0\Theta(A)_0 = A_0.

    Remember that A 0A_0 itself is a module over =( 1 0)\mathcal{H} = (\mathcal{H}_1 \stackrel{\to}{\to} \mathcal{H}_0), so that A 0= p 0(A 0) pA_0 = \coprod_{p \in \mathcal{H}_0} (A_0)_p.

  • For x(A 0) px \in (A_0)_p and y(A 0) qy \in (A_0)_q a morphism in Θ(A) 1\Theta(A)_1 from xx to yy is labeled by h 1h \in \mathcal{H}_1 and a(A 1) qa \in (A_1)_q

    x(h,a)(y=ρ(h)(x)a), x \stackrel{(h,a)}{\to} (y = \rho(h)(x) - \partial a) \,,

    where ρ\rho denotes the action of \mathcal{H} on A 0A_0.

    The composition law is given by

    y (h 1,a 1) (h 2,a 2) x (h 1h 2,ρ(h 2)(a 1)+a 2) z. \array{ && y \\ & {}^{\mathllap{(h_1, a_1)}}\nearrow && \searrow^{\mathrlap{(h_2,a_2)}} \\ x &&\stackrel{(h_1 \circ h_2, \rho(h_2)(a_1) + a_2)}{\to}&& z } \,.
  • For k2k \geq 2 the family of groups Θ(A) k\Theta(A)_k is over x(A 0) px \in (A_0)_p the group (A k) q(A_k)_q

    Θ(A) k2= p 0 x(A 0) q(A k)q \Theta(A)_{k \geq 2} = \coprod_{p \in \mathcal{H}_0} \coprod_{x\in (A_0)_q} (A_k)q
  • The boundary maps and actions are the obvious ones


(ordinary abelian chain complex as crossed complex)

Let C C_\bullet be an ordinary chain complex of abelian groups, i.e. a chain complex of modules over the trivial groupoid.

Then (ΘC) 1(\Theta C)_1 is the groupoid with objects C 0C_0 and morphisms {xb(x+b)}\{x \stackrel{b}{\to} (x + \partial b)\}. And for n2n \geq 2 we have that (ΘC) n(\Theta C)_n is xC 0C n\coprod_{x \in C_0} C_n.


These form a pair of adjoint functors

(Θ):ChnΘCrs (\nabla \dashv \Theta) : Chn \stackrel{\overset{\nabla}{\leftarrow}}{\underset{\Theta}{\to}} Crs


This is proposition 7.4.29.


8 The Higher Homotopy van Kampen Theorem and its applications

8.4 The chain complex of a filtered space and of a CW-complex

Let all topological spaces XX in the following by Hausdorff spaces that admit a universal cover. X^\hat X.


(homology chain complex of a filtered space)

For X=(X )X = (X_\bullet) a filtered space define a chain complex of modules over a groupoid 𝒞 (X)\mathcal{C}_\bullet(X) as follows.

The groupoid 𝒢:=Π 1(X,X 0)\mathcal{G} := \Pi_1(X,X_0) is the full subgroupoid of the fundamental groupoid of XX on points in X 0X_0.

For x 0Xx_0 \in X let X^(x 0):= yΠ 1(y,x 0)\hat X(x_0) := \coprod_y \Pi_1(y,x_0) be the standard model for the universal cover of XX in terms of homotopy classes of paths into x 0x_0.

For all xX 0=Obj(Π 1(X,X 0))x \in X_0 = Obj(\Pi_1(X,X_0)) take the modules over Π 1(X,X 0)\Pi_1(X,X_0) to be the relative homology group?s

(𝒞 0X) x:=H 0(X^ 0(x)) (\mathcal{C}_0 X)_x := H_0(\hat X_0(x))

and for n1n \geq 1

(𝒞 nX) x:=H n(X^ n(x),X^ n1(x)). (\mathcal{C}_n X)_x := H_n(\hat X_n(x),\hat X_{n-1}(x) ) \,.

The action of Π 1(X,X 0)\Pi_1(X,X_0) on this is the evident one induced by composition of paths.

This extends to a functor

𝒞 :FTopChn. \mathcal{C}_\bullet : FTop \to Chn \,.

This is def 8.4.1

The next proposition asserts that this notion of chain complex of a filtered topological space is reproduced by the combination of


If the filtered space X X_\bullet is connected then there is a natural isomorphism

𝒞 XΠX. \mathcal{C}_\bullet X \simeq \nabla \Pi X \,.

This is proposition 8.4.2 . Use the relative Hurewicz theorem to translate from homotopy groups to homology groups.

Example: Chains on the nn-simplex

(chains on the nn-simplex)

Consider X=Δ nX = \Delta^n, the standard topological nn-simplex regarded as a filtered space with the union of its kk-faces in degree kk.

Notice that since Δ n\Delta^n is a simply connected space in this case we have that for each basepoint x(Δ n) 0x \in (\Delta^n)_0 the universal cover X^ x=X\hat X_{x} = X coincices with XX.

We have that

𝒞 Δ nΠΔ nN Δ[n] \mathcal{C}_\bullet \Delta^n \simeq \nabla \Pi \Delta^n \simeq N_\bullet \Delta[n]

is, over each vertex x(Δ n) 0x \in (\Delta^n)_0, the normalized chain complex of chains on the simplicial set Δ[n]\Delta[n]

𝒞 0Δ n= n+1(𝒞 0Δ n) x= \mathcal{C}_0 \Delta^n = \mathbb{Z}^{n+1} \;\;\;\;\; (\mathcal{C}_0 \Delta^n)_x = \mathbb{Z}
(𝒞 1Δ n) x= n (\mathcal{C}_1 \Delta^n)_{x} = \mathbb{Z}^n


(𝒞 nΔ n) x=. (\mathcal{C}_n \Delta^n)_x = \mathbb{Z} \,.

We have moreover that Π 1(Δ n,(Δ n) 0)\Pi_1(\Delta^n, (\Delta^n)_0) is the codiscrete groupoid on n+1n+1 objects. It acts on the 𝒞 k(Δ n)\mathcal{C}_k(\Delta^n) by identity maps

(x ix j):(𝒞 kΔ n) x i=(𝒞 kΔ n) x j. (x_i \to x_j) : (\mathcal{C}_{k} \Delta^n)_{x_i} \stackrel{=}{\to} (\mathcal{C}_{k} \Delta^n)_{x_j} \,.

It follows in particular that for D D_\bullet an ordinary chain complex of abelian groups regarded as a complex of modules over a groupoid in the trivial way, morphisms of modules over groupoids

𝒞 Δ nD \mathcal{C}_\bullet \Delta^n \to D

are canonically identified with morphisms of ordinary chain complexes of abelian groups

N Δ[n]D. N_\bullet \Delta[n] \to D \,.

For more on this see Dold-Kan map and omega-nerve.

9 Tensor products and homotopies of crossed complexes

9.9 The homotopy addition lemma for a simplex

For Δ n\Delta^n the topological nn-simplex regarded as a filtered space in the canonical way, the fundamental crossed complex ΠX n\Pi X^n is a groupoid-version of the nn-oriental: the free strict ω-groupoid on a single nn-simplex.

9.10 Simplicial sets and crossed complexes

By the discussion at The homotopy addition lemma for a simplex the fundamental crossed complex ΠΔ n\Pi \Delta^n plays the role of the free strict nn-groupoid on the nn-simplex.

The cosimplicial \infty-groupoid

ΠΔ :ΔCrsStrGrpd \Pi \Delta^\bullet : \Delta \to Crs \simeq Str \infty Grpd

induced by the discussion at nerve and realization a simplicial nerve operation on strict ω-groupoid – an ω-nerve:


(simplicial nerve)

Let CC be a crossed complex. Its simplicial nerve N ΔCN^\Delta C \in sSet is

(N ΔC) n:=Crs(ΠΔ n,C) (N^\Delta C)_n := Crs(\Pi \Delta^n, C)

This is definition 9.10.2.

Remark 9.10.6 (Dold-Kan map and ω\omega-nerve)

(Dold-Kan map)

For DChnD \in Chn a chain complex (of abelian groups) regarded as a chain complex of modules over the trivial groupoid, we may regard it as a crossed complex ΘD\Theta D as described at Crossed complex from chain complex, hence as a strict ω-groupoid.

The ω-nerve N ΔΘDN^\Delta \Theta D \in sSet (described in Crossed complexes and simplicial sets) of this is the Kan complex underlying the image of DD under the Dold-Kan correspondence ChnsAbChn \to sAb.


By definition we have

N Δ(ΘD):=Crs(ΠΔ ,ΘD). N^\Delta (\Theta D) := Crs(\Pi \Delta^\bullet, \Theta D) \,.

By adjunction (ΠΘ)(\Pi \dashv \Theta) with the Theta-map this is equivalently

Chn(ΠΔ ,D) \cdots \simeq Chn( \nabla \Pi \Delta^\bullet, D)

Using the propositions and examples discussed at Chain complex of a filtered space we have that ΠΔ n\nabla \Pi \Delta^n is standard normalized chain complex N Δ[n]N_\bullet \Delta[n] of chains on the simplicial nn-simplex as discussed at chains on a simplicial set and Dold-Kan correspondence, but regarded as a complex of modules over the groupoid Π 1(Δ n,(Δ n) 0)\Pi_1(\Delta^n, (\Delta^n)_0). But since the groupoid action on DD is trivial, the above is equivalent to

Chn(N Δ ,D). \cdots \simeq Chn( N_\bullet \Delta^\bullet , D) \,.

This appears as remark 9.10.6 together with its footnote 116 .


In the cosmic cube of higher category theory this realizes two edges

ChainCplx Θ CrossedCplx N Δ KanCplx StrAbStrGrpd StrGrpd Grpd \array{ ChainCplx &\stackrel{\Theta}{\hookrightarrow}& CrossedCplx &\stackrel{N^\Delta}{\hookrightarrow}& KanCplx \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ StrAb Str\infty Grpd &\hookrightarrow& Str \infty Grpd &\hookrightarrow& \infty Grpd }

including strict ∞-groupoids with strict abelian ∞-group-structure – modeled as chain complexes of abelian groups – into strict ∞-groupoids – modeled as crossed complexes – into all ∞-groupoids – modeled as Kan complexes. The composite is the map Ch sAbKanCplxCh_\bullet \to sAb \to KanCplx to simplicial groups from the Dold-Kan correspondence.

10 Resolutions

11 The cubical classifying space of a crossed complex

12 Nonabelian cohomology: spaces, groupoids

Cubical ω\omega-groupoids

13 The algebra of crossed complexes and cubical ω\omega-groupoids

14 Cubical homotopy groupoid

14.8 The cubical Dold-Kan theorem

See also Dold-Kan correspondence.


For an extensive list of relevant publications see

Some selected references are:

  1. Olum, P., Non-abelian cohomology and van Kampen’s theorem, Ann. Math. 68 (1958) 658–667.

  2. Brown, R., On a method of P. Olum, J. London Math. Soc. 40 (1965) 303–304.

  3. Brown, R., Elements of Modern Topology, McGraw Hill, Maidenhead, 1968.

  4. Brown, R., Topology and Groupoids, Booksurge, 2006.

  5. Higgins, P.J., Presentations of groupoids, with applications to groups, Proc. Camb. Phil. Soc., 60 (1964) 7–20.

  6. Brown, R. and Higgins, P.J., On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc.(3) 36 (1978) 193–212.

  7. Brown, R. and Higgins, P.J., Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981) 11–41.

  8. Whitehead, J.H.C., Combinatorial Homotopy II, Bull. Amer. Math. Soc., 55 (1949), 453–496.

  9. Brown, R. Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems, in Handbook of Algebra 6, Edited M. Hazewinkel, Elsevier, 2009.

  10. Wensley, C.D. and Alp, M., XMOD, a GAP share package for computation with crossed modules, GAP Manual, (1997), 1355–1420.

  11. Brown, R., Higgins, P.J., and Sivera, R., Nonabelian Algebraic Topology: Filtered spaces, Crossed Complexes, Cubical Homotopy Groupoids, EMS Tracts in Mathematics, Vol. 15, (Autumn 2010).

category: reference

Revised on March 16, 2013 15:51:00 by Tim Porter (