CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
While the idea of a homotopy type is very suitable for the study of (locally) good topological spaces, the weak homotopy type fails to give useful information for ‘bad’ spaces, of which classical examples include the Warsaw Circle, Sierpinski gasket, p-adic solenoid and so on. Even if our initial and principal interest is more often than not in good spaces, bad spaces arise naturally in their study. For example, in the study of dynamical systems on manifolds, an important issue is the study of the attractors of such systems, which are typically fractal sets, and thus not ‘locally nice’ at all! The intuitive idea of shape theory is to define invariants of quite general topological spaces by approximating them with ‘good’ spaces, either by embedding them into good spaces, and looking at open or polyhedral neighborhoods of them, or by considering abstract inverse systems of good spaces. The two approaches are closely related.
If there are few maps from polyhedra (e.g. from spheres) into the space, then the weak homotopy type may tell too little about the space. Therefore one “expands” the space into a successive system of spaces which are good recipients of maps from polyhedra (e.g. ANR-s, polyhedra) and one adapts the homotopy theory to such expansions. The analogue of (strong) homotopy type in this setting is the shape of a space; the shape is an invariant of the strong homotopy type and agrees with it on the ANR-s for metric spaces and on the polyhedra. It is more crude for other spaces, but more suitable than the weak homotopy type, or more exactly gives complementary information. Instead of embedding a space, one may abstractly expand or resolve the space or its homotopy class into a pro-object in a category of nice spaces. Strong shape theory is a variant which is closer to the usual kind of homotopy, is more geometric and has more homotopy theoretic constructions available in its ‘toolkit’. It differs by passing to the homotopy category at a later stage in the theory, so one gets homotopy coherent approximating systems rather than homotopy commutative ones.
Shape theory was first explicitly introduced by Polish mathematician Karol Borsuk in the 1960s, although Christie, a student of Lefshetz, had done some initial development work on the same basic idea much earlier. One of the modern versions of shape theory is developed in terms of inverse systems of absolute neighbourhood retracts (ANRs) (which are pro-objects in the homotopy category of polyhedra). These were introduced in this setting by S. Mardešić, and J. Segal (1971) and independently, in a slightly different form, by Tim Porter (thesis, 1971), using the more combinatorial framework of pro-objects in the category of simplicial sets. This latter approach also indicated the possible link with the étale homotopy theory of Artin and Mazur, (Springer Lecture Notes 100).
Shape theory is a ‘Čech homotopy theory’, having a similar relationship to Čech homology as homotopy theory, based on the singular complex construction, has to singular homology. In fact, as mentioned above, the origins of both shape theory and strong shape theory go back than further Borsuk’s initial papers to work by Lefshetz and his student, D. Christie (thesis plus article, D.E. Christie, Net homotopy for compacta, Trans. Amer. Math. Soc., 56 (1944) 275–308). Christie considered a 2-truncated form of strong shape theory, categorically this corresponds to a lax or op-lax 2-categorical version of shape theory. Although many of the initial ideas were developed by Christie, the paper went unnoticed until Borsuk developed his slightly different approach in the late 1960s.
For many applications one needs more refined invariants which build up strong shape theory, while sometimes more crude versions may be useful, for example the recent theory of coarse shape?.
Strong Shape Theory developed in the 1970s through the work of Edwards and Hastings (lecture notes, see below), Porter, Quigley, and others. It has, especially in the approach pioneered by Edwards and Hastings, strong links to proper homotopy theory. The links are a form of duality related to some of the more geometric duality theorems of classical cohomology.
M. Batanin further elucidated strong shape theory from a categorical and 2-categorical point of view, but his approach is as yet not much used. His 1997 paper, shows the connections between this theory and a homotopy theory of simplicial distributors linked to $A_{\infty}$-categories.
The structure of the strong shape theory of compact spaces is related to certain structure and constructions on the corresponding (commutative) $C^*$-algebras of functions. These are related to the algebraic K-theory of such commutative $C^*$-algebras. Extensions to non-commutative $C^*$-algebras have been made; see (Blackadar) and (Dadarkat) below, for a start.
As shape theory is a Čech homotopy theory, its corresponding homology is Cech homology, but what is the corresponding construction for strong shape? The answer is Steenrod–Sitnikov homology. This is discussed in Mardešić’s book, Strong Shape and Homology, (see below). Many of the themes of homotopy coherence and related ideas occur in this theory and this suggests an infinity categorical approach (closely related to Batanin’s) may be important. This seems to be emerging with interpretations of work by Toen and Vezzosi, and by Lurie, and perhaps suggests a review of Batanin’s work from that new viewpoint.
This was the original form and applies to compact metric spaces. It uses the fact that any compact metric space can be embedded in the Hilbert Cube. For any such embedded compact metric spaces, $X$ and $Y$, one considers shape maps from the collection of open neighbourhoods of $X$ to those of $Y$. These shape maps are families of continuous maps satisfying a compatibility relationship ‘ up to homotopy’. These compose nicely and form the Borsuk shape category. Two spaces have the same shape if they are isomorphic in this category. Full details of the definition of such shape morphisms are given in the separate entry, Borsuk shape theory.
A remarkable and beautiful theorem of Chapman (the Chapman complement theorem) shows that the shape of two compact metric spaces, $X$ and $Y$ embedded in the pseudo-interior of the Hilbert cube, $Q$, have the same shape if and only if their complements $Q\setminus X$ and $Q\setminus Y$ are homeomorphic.
The idea of abstract shape theory is very simple. You have a category, $C$, of objects that you want to study. (In Borsuk’s classical topological case this was the (homotopy) category of compact metric spaces.) You have a well behaved set of methods that work well for some subcategory, $D$, of those objects (polyhedra in Borsuk’s case, where the methods were those of homotopy theory). The categorical idea that can be glimpsed behind the topological constructions of topological shape theory is that of replacing an object $X$ of $C$ with approximations to $X$ by objects of $D$, (so ‘approximating’ a compact metric space by polyhedra, for instance). Categorically this replaces the object $X$ by the comma category, $(X/D)$, which comes with a projection functor to $D$, which ‘records’ the approximating $D$-object for each approximation. You then use your invariants for objects in $D$ to define (and study) the more general objects in $C$. This does not come without consequences as you obtain new types of maps, (shape maps) between the objects of $C$, namely functors between the comma categories that respect the projections. The objects of $C$ together with your new shape maps form the shape category of your situation.
The shape category $Shape(C,D)$ is associated to a pair $(C,D)$ of a category $C$ and a dense subcategory $D$.
Here dense subcategory is used in the second sense of that term: for every object $X$ in $C$ there is its $D$-expansion, which is the object $\bar{X}$ in the category $pro D$ of pro-objects in $D$ that is universal (initial) with the property that it is equipped with a morphism $X\to\bar{X}$ in $pro D$.
The shape category $Shape(C,D)$ has
the same objects as $C$
its morphisms are equivalence classes of maps between the $D$-expansions.
A more categorical form of shape theory was studied by Deleanu and Hilton in a series of papers in the 1970s. They consider a more general setting of a functor $K : D \to C$, which in the classical Borsuk case would be the inclusion of the homotopy category of compact polyhedra into that of all compact metric spaces.
This was developed further by Bourn and Cordier, and a strong shape version was then found by Batanin.
The classical application of shape theoretic idea is to the study of topological spaces that do not have the homotopy type of a CW-complex. This is the case obtained from the above general setup by choosing
$C =$ HoTop${}_{he}$ the homotopy category of the category Top of all topological spaces localised at the homotopy equivalences;
$D = Ho(CW Cplx)$ the full subcategory given by CW-complexes.
More on this is in the section Shape theory for topological spaces below and in Cech homotopy.
Consider the category $C =$Grp of groups and its subcategory $D$ of finite group. A shape map between two groups is a map between their profinite completions. This sort of behaviour is quite general as this form of abstract shape theory is related to equational completions; see
This aspect is explored reasonably fully in the book by Cordier and Porter (see below).
A different terminology and slightly different emphasis is often used within the shape theoretic literature as it corresponds more to the geometric intuition needed there, deriving originally from the important classical motivation of Borsuk, Mardešić, and Segal.
…
There is a way to study the strong shape theory of a topological space $X$ in terms of ∞-stacks on $X$, i.e. in terms of the (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ on the category of open subsets of $X$. This is described in
and in section 7.1.6 of
For more details see shape of an (infinity,1)-topos.
This theory fits into the general picture above of a subcategory $D\subset C$, where now $C$ is the $(\infty,1)$-category of $(\infty,1)$-toposes, while $D$ is the category of $\infty$-groupoids, regarded as their presheaf $(\infty,1)$-toposes. Thus, the “shape” of an $(\infty,1)$-topos $X$ is the functor $Hom(X,-)\colon \infty Gpd \to \infty Gpd$.
Alternately, since a geometric morphism from an $(\infty,1)$-topos $X$ into presheaves on an $\infty$-groupoid $K$ is the same as a global section of the constant ∞-stack $L Const(K)$ over $X$, we can also describe this functor as the composite
Thus, we can equivalently describe the shape of $X$ by mapping out of it into topological spaces over $X$ that are at least fiberwise nice topological spaces: in other words, to look at $\infty$-covering spaces over $X$.
Now, for a small (∞,1)-category $C$, a functor $C \to \infty Grpd$ that preserves finite limits may be thought of as a pro-object in $C$. Now $\infty Gpd$ is not small, but one may hope that the functors $Shape(X)\colon \infty Gpd \to \infty Gpd$ arising in this way are determined by a small amount of data, and thus give honest pro-$\infty$-groupoids.
We can, if we wish, define for the nonce
to be the fully subcategory of (∞,1)-functors that preserve finite limits, although as discussed above this is not quite correct. We call the objects in $Pro(\infty Grpd)$ pro-spaces or shapes. Notice that by the homotopy hypothesis-theorem, we can think here of $\infty Grpd \simeq Top_{cg,wH}$ as the category of nice topological spaces, considered up to homotopy equivalence.
The first description of shapes makes it obviously functorial in geometric morphisms of $(\infty,1)$-toposes. This can be seen from the second definition as well: given $(f^* \dashv f_*) \colon \mathbf{H} \to \mathbf{K}$, the unit $Id_{\mathbf{K}} \to f_* \circ f^*$ induces a transformation
that may be regarded as a morphism of shapes
We say the geometric morphism $f$ is a shape invariance if $Shape(f)$ is an equivalence of pro-spaces.
For $f : X \to Y$ a continuous map of paracompact spaces, the induced geometric morphism $(f^* \dashv f*) : Sh_{(\infty,1)}(X) \to Sh_{(\infty,1)}(Y)$ is a shape equivalence, precisely if for each CW-complex $K$ the map
is an equivalence.
This is HTT, prop. 7.1.6.8.
In a dynamical system, the attractor?s are rarely polyhedra and their homotopy properties correspond more nearly to shape theoretic ones than to standard homotopy theoretic ones. This seems first to have been studied by Hastings? in 1988, (see references) and more recently has been explored in papers by José Sanjurjo and his coworkers, see below. Adapting this idea Hernandez, Teresa Rivas and José García Calcines? have been using ideas developed for proper homotopy theory and shape involving pro-spaces, to describe limiting properties of dynamical systems.
See also $n$lab entries shape fibration, approximate fibration, … and references
The original references for the shape theory of metric compacta are:
The ‘ANR-systems’ approach of Mardešić and Segal appeared in
and is fully developed in
The more or less equivalent pro-object approach was independently developed by Porter in
References relating more to strong shape theory include:
D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag, pdf
J.T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape theory, Glasnik Mat. 19(39) (1984) 335–399.
S. Mardešić, Strong shape and homology, Springer monographs in mathematics, Springer-Verlag.
Michael Batanin, Categorical strong shape theory, Cahiers Topologie Géom. Différentielle Catég. 38 (1997), no. 1, 3–66.
T. Porter, Stability Results for Topological Spaces, Math. Zeit. 150, 1974, pp. 1-21.
T. Porter, Abstract homotopy theory in procategories, Cahiers Top. Géom. Diff., 17, 1976, pp. 113-124, numdam
T. Porter, Coherent prohomotopical algebra, numdam, Cahiers Top. Géom. Diff. 18, (1978) pp. 139-179;
T. Porter, Coherent prohomotopy theory, Cahiers Top. Géom. Diff. 19, (1978) pp. 3-46, numdam.
These last three papers developed a version of the BrownAHT to pro-categories of simplicial sets and of chain complexes, so as to give strong shape theory a better foundation and toolbox of homotopical methods. These methods were complementary to those of Edwards and Hastings, (listed above), who used a Quillen model category structure on the pro-category.
References to the categorical forms of shape theory include
A. Deleanu, P.J. Hilton, On the categorical shape of a functor, Fund. Math. 97 (1977) 157 - 176.
A. Deleanu, P.J. Hilton, Borsuk’s shape and Grothendieck categories of pro-objects, Math. Proc. Camb. Phil. Soc. 79 (1976) 473-482.
D. Bourn, J.-M. Cordier, Distributeurs et théorie de la forme, Cahiers Topologie Géom. Différentielle Catég. 21,(1980), no. 2, 161–188, numdam.
and
which explores categorical methods in the area.
The relationship between invariants of $C^*$-algebras and the shape of their spectra was explored in
The links are with K-theory and Kasparov’s theory. This connection and a related one to ‘asymptotic morphisms’ is explored in some neat notes by Anderson and Grodal:
That connection with asymptotic morphisms is fully explored in the work of Dadarlat; see his papers,
For links with dynamical systems, see the early paper,
and more recently
Joel W. Robbin, Dietmar A. Salamon, Dynamical systems, Shape Theory and the Conley index, Ergodic Theory Dynam. Systems 8 (1988) 375 - 393,
A. Giraldo, M. A. Morón, F. R. Ruiz del Portal, J. M. R. Sanjurjo, Shape of global attractors in topological spaces, Nonlinear Analysis 60 (2005) 837 - 847
and