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 (2011)
ISBN 978-3-03719-083-8
This treats algebraic topology using tools of strict ∞-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. The philosophy behind the work is also given in the paper Modelling and Computing Homotopy Types: I
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 $n \gt 1$.
It took 9 years to find such a gadget in dimension $2$, 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 $\pi_3$ 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 $\rho_2(X,A)$, with its compositions in $2$ 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 ∞-groupoid? $\rho X_*$ of a filtered space $X_*$, and find its relation to the quite classical homotopically defined fundamental crossed complex functor $\Pi: (filtered spaces) \to (crossed complexes)$. The proofs here are non trivial. By proving using $\rho$ a colimit theorem for $\Pi$ 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 $n$-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!
A module over a groupoid is a collection of abelian groups equipped with a linear action by a groupoid.
(module over a groupoid)
Let $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)$ be a groupoid. A module over the groupoid $\mathcal{G}$ is a collection $\{N_x\}_{x \in \mathcal{G}_0}$ of abelian groups equipped with a collection of maps
that are linear and respect the groupoid composition in the obvious way.
(…)
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 $X_*$ one can consider the variant where the k-morphisms of the fundamental $\infty$-groupoid are constrained to lie in $X_k$. 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 $X_\bullet$ be a filtered space.
Write $\Pi_1(X_1,X_0)$ for the subgroupoid of the fundamental groupoid $\Pi_1(X_1)$ of $X_1$ on objects that are in $X_0$.
The fundamental crossed complex $\Pi X_*$ of $X$ is the crossed complex
with
and
where $\pi_n(X_n, X_{n-1}, x)$ is the relative homotopy group? obtained by equivalence classes of maps from the pointed $n$-disk into $X$ such that the disk lands in $X_n$, its boundary in $X_{n-1}$ and its basepoint on $x$.
See also section 1 of
(fundamental crossed complex of the $n$-simplex)
The topological $n$-simplex $\Delta^n$ is canonically a filtered space with $(\Delta^n)_k$ being the union of its $k$-faces.
Then we have that $\Pi_1((\Delta^n)_1, (\Delta^n)_0)$ is the groupoid whose objects are the $n+1$ vertices of $\Delta^n$ and which has precisely one morphism $x_i \to x_j$ for each ordered pair $x_i,x_j \in (\Delta^n)_0$ (all of them being isomorphisms)
At any $x_i$ the relative homotopy group $\pi_2((\Delta^n)_2,(\Delta^n)_1, x_i)$ is a group on the set of 2-faces that have $x_i$ as a 0-face: there is a unique homotopy class of disks in $\Delta^n$ that sits in the 2-faces $(\Delta^n)_2$, whose base point is at $x_j$ and whose boundary runs along the boundary of a given 2-face of $\Delta^n$.
So (using the equivalence of crossed complexes with strict $\omega$-groupoids) for instance $\Pi \Delta^2$ is generated from $\Pi_1((\Delta^2)_1,(\Delta^2)_0)$ as above and a 2-cell
under whiskering and composition. For instance whiskering this with $x_1 \to x_2$ yields the 2-morphism
One sees that $\Pi \Delta^2$ is the strict groupoidification of the second oriental.
Generally, $\Pi \Delta^n$ is the $n$-groupoid freely generated from $k$-morphisms for each $k$-face of $\Delta^n$.
(groupoid module chain complexes)
Write $Chn$ for the category of chain complexes of modules over a groupoid.
This is Def. 7.4.1.
Given a module over a groupoid $(N,\mathcal{G})$, the semidirect product groupoid $\mathcal{G} \ltimes N$ has the same objects as $\mathcal{G}$ and morphisms
with composition given by the action of $\mathcal{G}$ on $N$.
This is def. 7.4.5
(covering morphism)
For $(t : N \to \mathcal{G}_0,\mathcal{G})$ a module over a groupoid, write $P(N,\mathcal{G})$ for the groupoid $\mathcal{G}$ pulled back to the underlying set of $N$:
an object of $P(N,\mathcal{G})$ is an element in $N$ and a morphism $n_1 \to n_2$ is a morphism $\mathcal{G}(t(n_1),t(n_2))$.
This is def. 7.4.9.
(chain complex from a crossed complex)
Define a functor $\nabla : Crs \to Chn$ from crossed complexes to modules over groupoids as follows:
For $C$ a crossed complex we set for $n \geq 3$
and for $n \leq 2$ it is given by …
This is definition 7.4.20.
We describe a construction of a crossed complex from a chain complex of modules over a groupoid $(A_n, \mathcal{H})$. 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 ∞-groupoids.
Recall the definition of the semidirect product groupoid $\mathcal{H} \ltimes A_n$.
(crossed complex from a chain complex)
For $A$ a chain complex of modules over a groupoid $\mathcal{H}$, let $\Theta A \in Crs$ be the crossed complex
where
and where
is the canonical covering morphism from above.
Here $\mathcal{H} \ltimes A_1$ acts on $A_n$ for $n \geq 2$ via the projection $\mathcal{H} \ltimes A_1 \to \mathcal{H}$, i.e. $A_1$ acts trivially. (…)
Finally set $\Theta(A)_0 := A_0$.
We spell out what this boils down to explicitly.
Explicit description
Let $A_\bullet$ be a chain complex of modules over the groupoid $\mathcal{H}$. Then the crossed complex $\Theta(A)$ is the following.
Its set of objects is $\Theta(A)_0 = A_0$.
Remember that $A_0$ itself is a module over $\mathcal{H} = (\mathcal{H}_1 \stackrel{\to}{\to} \mathcal{H}_0)$, so that $A_0 = \coprod_{p \in \mathcal{H}_0} (A_0)_p$.
For $x \in (A_0)_p$ and $y \in (A_0)_q$ a morphism in $\Theta(A)_1$ from $x$ to $y$ is labeled by $h \in \mathcal{H}_1$ and $a \in (A_1)_q$
where $\rho$ denotes the action of $\mathcal{H}$ on $A_0$.
The composition law is given by
For $k \geq 2$ the family of groups $\Theta(A)_k$ is over $x \in (A_0)_p$ the group $(A_k)_q$
The boundary maps and actions are the obvious ones…
(ordinary abelian chain complex as crossed complex)
Let $C_\bullet$ be an ordinary chain complex of abelian groups, i.e. a chain complex of modules over the trivial groupoid.
Then $(\Theta C)_1$ is the groupoid with objects $C_0$ and morphisms $\{x \stackrel{b}{\to} (x + \partial b)\}$. And for $n \geq 2$ we have that $(\Theta C)_n$ is $\coprod_{x \in C_0} C_n$.
These form a pair of adjoint functors
where…
This is proposition 7.4.29.
(…)
Let all topological spaces $X$ in the following by Hausdorff spaces that admit a universal cover. $\hat X$.
(homology chain complex of a filtered space)
For $X = (X_\bullet)$ a filtered space define a chain complex of modules over a groupoid $\mathcal{C}_\bullet(X)$ as follows.
The groupoid $\mathcal{G} := \Pi_1(X,X_0)$ is the full subgroupoid of the fundamental groupoid of $X$ on points in $X_0$.
For $x_0 \in X$ let $\hat X(x_0) := \coprod_y \Pi_1(y,x_0)$ be the standard model for the universal cover of $X$ in terms of homotopy classes of paths into $x_0$.
For all $x \in X_0 = Obj(\Pi_1(X,X_0))$ take the modules over $\Pi_1(X,X_0)$ to be the relative homology group?s
and for $n \geq 1$
The action of $\Pi_1(X,X_0)$ on this is the evident one induced by composition of paths.
This extends to a functor
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
the fundamental crossed complex $\Pi X_\bullet$
and the chain complex of a crossed complex $\nabla \Pi X_\bullet$.
If the filtered space $X_\bullet$ is connected then there is a natural isomorphism
This is proposition 8.4.2 . Use the relative Hurewicz theorem to translate from homotopy groups to homology groups.
(chains on the $n$-simplex)
Consider $X = \Delta^n$, the standard topological $n$-simplex regarded as a filtered space with the union of its $k$-faces in degree $k$.
Notice that since $\Delta^n$ is a simply connected space in this case we have that for each basepoint $x \in (\Delta^n)_0$ the universal cover $\hat X_{x} = X$ coincices with $X$.
We have that
is, over each vertex $x \in (\Delta^n)_0$, the normalized chain complex of chains on the simplicial set $\Delta[n]$
etc.
We have moreover that $\Pi_1(\Delta^n, (\Delta^n)_0)$ is the codiscrete groupoid on $n+1$ objects. It acts on the $\mathcal{C}_k(\Delta^n)$ by identity maps
It follows in particular that for $D_\bullet$ 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
are canonically identified with morphisms of ordinary chain complexes of abelian groups
For more on this see Dold-Kan map and omega-nerve.
For $\Delta^n$ the topological $n$-simplex regarded as a filtered space in the canonical way, the fundamental crossed complex $\Pi X^n$ is a groupoid-version of the $n$-oriental: the free strict ∞-groupoid on a single $n$-simplex.
By the discussion at The homotopy addition lemma for a simplex the fundamental crossed complex $\Pi \Delta^n$ plays the role of the free strict $n$-groupoid on the $n$-simplex.
The cosimplicial $\infty$-groupoid
induced by the discussion at nerve and realization a simplicial nerve operation on strict ∞-groupoid – an ∞-nerve:
(simplicial nerve)
Let $C$ be a crossed complex. Its simplicial nerve $N^\Delta C \in$ sSet is
This is definition 9.10.2.
(Dold-Kan map)
For $D \in Chn$ 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 $\Theta D$ as described at Crossed complex from chain complex, hence as a strict ∞-groupoid.
The ∞-nerve $N^\Delta \Theta D \in$ sSet (described in Crossed complexes and simplicial sets) of this is the Kan complex underlying the image of $D$ under the Dold-Kan correspondence $Chn \to sAb$.
By definition we have
By adjunction $(\Pi \dashv \Theta)$ with the Theta-map this is equivalently
Using the propositions and examples discussed at Chain complex of a filtered space we have that $\nabla \Pi \Delta^n$ is standard normalized chain complex $N_\bullet \Delta[n]$ of chains on the simplicial $n$-simplex as discussed at chains on a simplicial set and Dold-Kan correspondence, but regarded as a complex of modules over the groupoid $\Pi_1(\Delta^n, (\Delta^n)_0)$. But since the groupoid action on $D$ is trivial, the above is equivalent to
This appears as remark 9.10.6 together with its footnote 116 .
In the cosmic cube of higher category theory this realizes two edges
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 $Ch_\bullet \to sAb \to KanCplx$ to simplicial groups from the Dold-Kan correspondence.
See also Dold-Kan correspondence.
For an extensive list of relevant publications see
Some selected references are:
Paul Olum, Non-abelian cohomology and van Kampen’s theorem, Ann. Math. 68 (1958) 658–667.
Brown, R., On a method of P. Olum, J. London Math. Soc. 40 (1965) 303–304.
Brown, R., Elements of Modern Topology, McGraw Hill, Maidenhead, 1968.
Brown, R., Topology and Groupoids, Booksurge, 2006.
Higgins, P.J., Presentations of groupoids, with applications to groups, Proc. Camb. Phil. Soc., 60 (1964) 7–20.
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.
Brown, R. and Higgins, P.J., Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981) 11–41.
Whitehead, J.H.C., Combinatorial Homotopy II, Bull. Amer. Math. Soc., 55 (1949), 453–496.
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.
Wensley, C.D. and Alp, M., XMOD, a GAP share package for computation with crossed modules, GAP Manual, (1997), 1355–1420.
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).
Last revised on February 15, 2024 at 14:29:37. See the history of this page for a list of all contributions to it.