Theory and Applications of Categories, Vol. 18, No. 19, 2007, pp. 602–630.

COLLARED COSPANS, COHOMOTOPY AND TQFT (COSPANS IN ALGEBRAIC TOPOLOGY, II)

MARCO GRANDIS

Abstract.

Topological cospans and their concatenation, by pushout, appear in the theories of tangles, ribbons, cobordisms, etc. Various algebraic invariants have been introduced for their study, which it would be interesting to link with the standard tools of Algebraic Topology, (co)homotopy and (co)homology functors.

Here we introduce collarable (and collared) cospans between topological spaces. They generalise the cospans which appear in the previous theories, as a consequence of a classical theorem on manifolds with boundary. Their interest lies in the fact that their concatenation is realised by means of homotopy pushouts. Therefore, cohomotopy functors induce ‘functors’ from collarable cospans to spans of sets, providing - by linearisation

- topological quantum field theories (TQFT) on manifolds and their cobordisms. Similarly, (co)homology and homotopy functors take collarable cospans to relations of abelian groups or (co)spans of groups, yielding other ‘algebraic’ invariants.

This is the second paper in a series devoted to the study of cospans in Algebraic Topology. It is practically independent from the first, which deals with higher cubical cospans in abstract categories. The third article will proceed from both, studying cubical topological cospans and their collared version.

Introduction

A cospan of topological spaces is a pair of maps with the same codomain

u = (u^{−}: X^{−} → X^{0} ← X^{+} : u^{+}),  (1)

viewed as a kind of morphism u: X^{−} ·→ X^{+}; they are composed by pushouts, and - of course - this composition is not strictly associative. They form a bicategory [1] and, also, the weak arrows of a weak double category ℂosp(Top), whose strict arrows are the ordinary maps of topological spaces [11]. But it is well known that pushouts of spaces are not homotopy invariant, generally, and do not behave well under homology or homotopy functors.

Work supported by a research grant of Università di Genova. Received by the editors 2007-07-13. Transmitted by J.-L. Loday. Published on 2007-11-27. 2000 Mathematics Subject Classification: 18D05, 55Q55, 57N70, 57M27, 57R56. Key words and phrases: spans, cospans, bicategory, weak double category, homotopy pushout, coho-

motopy functors, cobordism, knots, links, tangles, topological quantum field theories.

\begin{matrix}c \\ ◯ \end{matrix} Marco Grandis, 2007. Permission to copy for private use granted.

Here, we are interested in particular cospans, the collarable ones (2.2), which give rise to a substructure ℂblc(Top) ⊂ ℂosp(Top). Typically, (non oriented) tangles, ribbons and cobordisms between manifolds (as studied in Topological Quantum Field Theories) are collarable cospans and are composed by means of pushouts (in the following figures, collars are suggested by dashes or dots)

[figure: figure 1; •; •; •; •]

The structural interest of restricting to collarable cospans abides in the fact that the ‘collarable pushouts’ appearing in their composition are always homotopy pushouts (Thm. 2.5), hence homotopy invariant and ‘respected’ by most homotopy and homology functors, in some sense specified below.

First, the Borsuk cohomotopy functor π^{S} = [−, S]: Top → Set^{op} (where S is an arbitrary topological space) always turns a collarable pushout into a weak pullback of sets, and extends thus to a functor

π^{S}: Cblc_{iso}(Top) → Rel(Set),  (2)

defined on the category of topological spaces and isomorphism-classes of collarable cospans, with values in the category of relations of sets (Thm. 3.3). Under suitable restrictions, we get an extension with values in Sp_{iso}(fSet), the category of isomorphism-classes of spans of finite sets; this extension can be ‘linearised’ with values in finite-dimensional vector spaces (or free modules on a given ring), providing various topological quantum field theories.

Similar extensions can be obtained for relative homology or cohomology (4.2, 4.4). The same holds for the fundamental-groupoid functor, with values in cospans of groupoids (4.5), as a consequence of R. Brown’s version of the Seifert-van Kampen theorem [5, 6]. (One can not use the fundamental group, as a pointed cospan can not admit collars.)

In Section 1 we recall the construction of the weak double category ℂosp(Top) of topological cospans. It contains the corresponding bicategory, which could also be used here; but we want to have general ‘transversal maps’ in the structure, which allows for limits (cf. 1.3) and adjunctions, as studied in [11, 12]. Moreover, this setting is adequate for the cubical extension which will be given in Part III, on the basis of the weak cubical category of higher cospans of Part I [10].

In Section 2 we define collarable cospans; they form a weak double subcategory

ℂblc(Top) ⊂ ℂosp(Top).

Notice that, as motivated in 1.6, the definition is more general than one might expect, and also the degenerate cospans e_{1}(X) = (idX, idX) are accepted: a collarable cospan decomposes into a sum of a trivially collarable part (a pair of homeomorphisms) and a 1-collarable part; only the second admits ‘real’ collars. We end with a notion of weak equivalence of topological cospans, which is sufficiently general to make the cylindrical cospan on a space equivalent to the degenerate one (2.8); homotopy invariance of functors on topological cospans is defined with respect to this notion. Collared cospans, which are equipped with collars, are only hinted at (see 2.2); they will be studied in Part III of this series, together with their higher cubical versions.

Sections 3 and 4 show that cohomotopy functors, (co)homology theories and the fundamental-groupoid functor can be extended to collarable cospans, as outlined above, obtaining homotopy-invariant (co)lax or pseudo double functors. Some Frobenius algebras derived from cohomotopy functors are computed in 3.4, 3.5. A few computations on (absolute or relative) homology relations can be found in 4.6, 4.7. Finally, the Appendix in Section 5 deals with extending functors to (co)spans and relations, by classifying the squares of ‘ordinary maps’ which become ‘bicommutative’ in these involutive categories.

As to literature, a clear, elementary exposition of low-dimensional TQFT can be found in [16]. Other invariants for links, tangles and ribbons can be seen, for instance, in [15, 2] and their references.

Acknowledgements. The author gratefully acknowledges helpful discussions with Renzo Piccinini, Timothy Porter and Marco Zunino, and various accurate suggestions by the referee.

1. Topological cospans and homotopy pushouts

We review here the basic notions of this paper: the weak double category of topological cospans, with its limits and colimits, and the classical notion of homotopy pushout. The index α always takes the values ±1, which are written −, + in superscripts. IX = X×[0, 1] denotes the cylinder on a space X.

1.1. Topological cospans and their transversal maps. The weak double category ℂosp(Top) of topological spaces, maps and cospans, has been introduced and studied in [11]. It is also the 1-truncated structure of the weak cubical category of higher cospans ℂosp_{∗}(Top) introduced in Part I of this series [10]: truncation acts on the weak directions, leaving only one of them (together with the strict direction); here we will use a terminology and notation similar to the one of [10], which is more suited for a future study of higher topological cospans.

Therefore, a 0-cube, or object, is a topological space, viewed as a functor X: 1 → Top defined on the singleton category 1 = {∗}. A 1-cube is a (topological) cospan, i.e. a pair of continuous mappings between topological spaces

u = (u^{−}: X^{−} → X^{0} ← X^{+} : u^{+}),  (3)

and is viewed as a functor u: ∧ → Top, defined on the category

∧ :  −1 → 0 ← 1  (the formal cospan).  (4)

(An n-cube in the higher cubical structure ℂosp_{∗}(Top) is a functor ∧^{n} → Top [10]). Ordinary cospans form a 1-truncated cubical set, or reflexive graph, with (cubical) faces and degeneracies

∂_{1}^{α}u = X^{α},  e_{1}(X) = (id: X → X ← X : id).  (5)

We often write u = (u^{−}, u^{+}): X^{−} · → X^{+} to specify the cubical faces of u.

A (transversal) 0-map f: X → Y is a continuous mapping between topological spaces, also viewed as a natural transformation f: X → Y : 1 → Top. A (transversal) 1-map f: u → v, or double cell, is a natural transformation f: u → v: ∧ → Top, which amounts to two commutative squares in Top

\xymatrix{X^{−} \ar[r]^{f^{−}} \ar[d]_{u^{−}} & Y^{−} \ar[d]^{v^{−}} \\ X^{0} \ar[r]^{f^{0}} & Y^{0} & f = (f^{−}, f^{0}, f^{+}): u → v. \\ X^{+} \ar[r]_{f^{+}} \ar[u]^{u^{+}} & Y^{+} \ar[u]_{v^{+}}}  (6)

Also transversal maps (of cubical degree 0 or 1) form a 1-truncated cubical set, with:

∂_{1}^{α}f = f^{α},  e_{1}(f) = (f, f, f)  (α = ±1).  (7)

The 1-map f is said to be special if its cubical faces f^{−}, f^{+} are identities.

Transversal maps compose (as natural transformations), forming the category of diagrams Top^{∧}, with identities id(u) = (idX^{−}, idX^{0}, idX^{+}). Cospans (and their transversal maps) are concatenated using pushouts: given u and v = (v^{−}: Y^{−} → Y^{0} ← Y^{+} : v^{+}), with X^{+} = A = Y^{−}, the concatenation w = u +_{1} v is computed as:

[figure: Z^{0}; X^{0}; Y^{0}; w = u +_{1} v = (X^{−} → Z^{0} ← Y^{+}).; X^{−}; A; Y^{+}]  (8)

This ‘cubical’ composition is categorical up to isomorphisms of cospans, i.e. invertible special transversal maps.

The weak double category structure is described more in detail in [11] (and [10]). In [11], the transversal (strict) direction is called horizontal and the cubical (weak) direction is called vertical. A 0-map is called a horizontal arrow, a 1-cube is a vertical arrow, a 1-map is always called a double cell. (In other articles on weak double categories, ‘horizontal’ and ‘vertical’ are interchanged, which agrees with the usual terminology of bicategories. Here, such terms will be avoided.)

We will also use the associated category Cosp_{iso}(Top), of topological spaces and isomorphism classes of cospans, up to invertible special 1-maps.

1.2. Remarks. The bicategory of topological cospans [1] can be identified to the weak double subcategory of ℂosp(Top) obtained by restricting 0-maps to identities of topological spaces and 1-maps to the special ones.

We prefer to use the larger structure of a weak double category, where (double) limits and colimits exist and essentially amount to (co)limits in the category of diagrams Top^{∧}, together with the (co)tabulator, i.e. the double (co)limit of a 1-cube (see [11]: the cotabulator of a cospan is its central space, while the tabulator is its pullback). The (co)limits of interest for the sequel are briefly described below (1.3).

Finally, let us remark that concatenation in ℂosp(Top) is well-defined once we assume that we have in Top (or any other setting for cospans) a (symmetric) choice of a distinguished pushout for any span. As discussed in [10], 3.1, it is convenient to assume the following restriction on the choice (which is formulated there in a more general context)

(a) the pair (f, 1) has distinguished pushout (1, f), and symmetrically (unitarity constraint).

1.3. Limits and colimits. The product ∏ u_{j} of a family of cospans u_{j}: ∧ → Top (j ∈ J) is a cospan u equipped with a family p_{j}: u → u_{j} of 1-maps satisfying the usual universal property; it is simply computed componentwise: X^{−} = ∏ X_{i}^{−}, and so on.

Similarly, a sum ∑ u_{j} of cospans is computed componentwise, and the (co)equaliser of two 1-maps f, g: u → v, in the category of diagrams Top^{∧}, consists of the three (co)equalisers of the components.

A sub-cospan of u = (u^{−}: X^{−} → X^{0} ← X^{+} : u^{+}) will be a regular subobject of u (an equaliser of two 1-maps, as above). Therefore, it amounts to assigning three subspaces (Y^{−}, Y^{0}, Y^{+}) such that

Y^{t} ⊂ X^{t},  u^{α}(Y^{α}) ⊂ Y^{0}  (t ∈ ∧; α = ±),  (9)

and we say that the sub-cospan is open (resp. closed) in u if so are the three subspaces Y^{t} ⊂ X^{t}. The sub-cospans of u form a complete lattice, which is a sublattice of P(X^{−})× P(X^{0})×P(X^{+}).

Finally, let us recall that decomposing a space X into a categorical sum X = ∑ X_{j} (in Top) amounts to giving a partition of the space X into a family of clopens (closed and open subspaces). Similarly, to give a decomposition u = ∑ u_{j} into a sum of cospans amounts to give a clopen partition (u_{j}) of u, i.e. a cover of u by disjoint sub-cospans, closed and open in u.

1.4. Standard homotopy pushouts. We recall now a fundamental notion of homotopy theory, introduced by Mather [18].

Let f: A → X, g: A → Y form a span in Top. The standard homotopy pushout from f to g is a four-tuple (P; u, v; λ) as in the left diagram below, where λ: uf → vg: A → P is a homotopy satisfying the following universal property (as for cocomma squares of categories), which determines it up to homeomorphism

[figure: A; g; d^{+}; Y; v; A; g; f; Y; v; A; d^{−}; f; IA; λ; X; u; λ; P; X; u; P]  (10)

(a) for every λ′: u′f → v′g: A → W, there is precisely one map h: P → W such that u′ = hu, v′ = hv, λ′ = hλ.

(Writing hλ we are using the obvious whisker composition of homotopies and maps). In Top, the solution always exists and can be constructed as the ordinary colimit of the right-hand diagram above. This construction is based on the cylinder IA = A×[0, 1] and its faces

d^{−}, d^{+} : A → IA,  d^{−}(a) = (a, 0),  d^{+}(a) = (a, 1) (a ∈ A).  (11)

Therefore, the space P is a pasting of X and Y with the cylinder IA, and can be realised as a quotient of their topological sum, under the equivalence relation which gives the following identifications:

P = (X + IA + Y)/∼,  [f(a)] = [a, 0],  [g(a)] = [a, 1] (a ∈ A).  (12)

As a crucial feature, this construction always has strong properties of homotopy invariance (e.g., see [9], Section 3), which an ordinary pushout - generally - does not have. Notice also that the cylinder IA is itself the standard homotopy pushout from idA to idA.

1.5. General homotopy pushouts. As in Mather’s original paper, a homotopy pushout of f, g will be any space P′ homotopy equivalent to the standard homotopy pushout P.

With the notation of 1.4, the triple (u, v, λ) of P, composed with an equivalence-map P → P′, gives maps u′: X → P′, v′: Y → P′ and a homotopy λ′: u′f → v′g: A → P′ satisfying the following weak universal property, of mere existence:

(a) for every λ′′: u′′f → v′′g: A → P′′, there is some map k: P′ → P′′ such that u′′ ≃ku′ and v′′ ≃kv′.

It follows easily that every homotopy pushout becomes a weak pushout in the homotopy category HoTop = Top/ ≃ (of spaces and homotopy classes of maps). Furthermore, every (contravariant) cohomotopy functor [−, S]: Top → Set takes a homotopy pushout of spaces to a weak pullback of sets, a fact which will be used below.

1.6. Degenerate and cylindrical cospans. Before going on to define collarable maps, in the next section, it will be useful to compare the degenerate cospan e_{1}(X) = (idX, idX) of a space X with the cylindrical cospan

E_{1}(X) = (d^{−}: X → IX ← X : d^{+}),  d^{−}(x) = (x, 0),  d^{+}(x) = (x, 1),  (13)

[figure: X; X; e_{1}(X); X; X; E_{1}(X)]

which are produced by the degenerate span X ← X → X, as an ordinary pushout and a standard homotopy pushout, respectively. (They coincide for X = ∅.)

Assuming that X ≠ ∅, let us note that these cospans are not isomorphic, i.e. cannot be linked by an invertible special transversal map. (They are just weakly equivalent cospans, as defined in 2.8).

In fact, we have a special transversal 1-map p: E_{1}(X) → e_{1}(X), as in the left diagram below

\xymatrix{X \ar[rr]^{d^{−}} \ar[dd]_{1} &  & IX \ar[dd]^{p} &  & X \ar[ll]_{d^{+}} \ar[dd]^{1} & X \ar[rr]^{1} \ar[dd]_{f^{−}} &  & X \ar[dd]^{f^{0}} &  & X \ar[ll]_{1} \ar[dd]^{f^{+}} \\  & = &  & = &  &  & ? &  & ? \\ X \ar[rr]_{1} &  & X &  & X \ar[ll]^{1} & X \ar[rr]_{d^{−}} &  & IX &  & X \ar[ll]^{d^{+}}}  (14)

but there is no transversal map f going backwards (special or not): in the right diagram above, any point x ∈ X would give (f^{−}(x), 0) = f^{0}(x) = (f^{+}(x), 1).

Now, the cylinder cospans admits ‘collars’ (suggested by dashed lines, in the figure above), while e_{1}(X) does not (for X ≠ ∅). However, working with the cylindrical cospans as weak identities for the collarable cospans gives various problems. In Part III, extending our construction to cubical collared cospans, of any degree, we will see that the cubical relation e_{1}e_{1} = e_{2}e_{1} cannot be fulfilled with cylindrical cospans: we would not even get a cubical set. Already here (as a minor point), if we want the new structure to be a weak double subcategory of ℂosp(Top), we must keep the same degeneracies. This is why we will use a generalised notion of collarable cospan, including the degenerate ones as ‘trivially collarable’.

Finally, one can think of a third alternative, which presents other technical problems. One can modify ℂosp(Top) with a ‘better’ concatenation of consecutive cospans, say u ⊗_{1} v, realised by means of standard homotopy pushouts (instead of using ordinary pushouts and restricting cospans, as we do below). Now, associativity works well for the new concatenation, up to invertible special transversal maps, as one can easily see. But degenerate cospans do not work as weak identities: for a space X, we get e_{1}(X) ⊗_{1} e_{1}(X) = E_{1}(X), which is not isomorphic to e_{1}(X). On the other hand, we have already seen that, using the cylindrical cospans instead of the degenerate ones, as weak identities, gives other problems in the higher cubical structure.

2. Collarable cospans and collarable pushouts

We introduce here the weak double subcategory of collarable cospans ℂblc(Top) ⊂ ℂosp(Top). They are concatenated by means of pushouts of collarable maps, which are homotopy pushouts (Thm. 2.5). This will be crucial for applying cohomotopy functors (Section 3), or (co)homology and homotopy functors (Section 4).

2.1. Collarable maps. The idea of a topological cospan which ‘admits collars’, inspired by cobordisms between manifolds, is rather intuitive. Examples can be seen in fig. 1 of the Introduction, in the right-hand figure of (13) and, below, in all the figures of (44), (75), (77). We have already seen above that it is convenient to generalise it, to include degenerate cospans (1.6); similarly, we must use a generalised notion of collarable map.

Let us begin by recalling that a closed injective map f: X → Y in Top is always a topological embedding, in the sense that X has the pre-image topology (the less fine one which makes f continuous): indeed, if C is closed in X, then C = f^{−1}(f(C)) is the pre-image of a closed subset of Y. Such maps will be called closed (topological) embeddings.

We say that a map f: X → Y between topological spaces:

(i) is 0-collarable, or trivially collarable, if it is a homeomorphism, (ii) is 1-collarable, or admits a collar, if it has a (continuous) extension F to the cylinder IX, which is a closed embedding

F: IX → Y,  f = F (−, 0): X → Y,  (15)

so that the subset F (X ×[0, 1[ ) is open in Y.

The only map which is both 0- and 1-collarable is id(∅) (because F is injective and F (X ×I) ⊂ Y = f(X) = F (X ×{0})).

Now, a collarable map f: X → Y will be a continuous mapping which can be decomposed in a sum of two maps:

f = f_{0} + f_{1} : (X_{0} + X_{1}) → (Y_{0} + Y_{1})  (collarable decomposition),  (16)

where f_{0} is 0-collarable and f_{1} is 1-collarable.

Each component f_{i} and f itself are closed embeddings. The collarable decomposition of f is uniquely determined (if it exists): given a second, based on the decomposition X = X′_{0} + X′_{1}, f must be both 0- and 1-collarable on the subspaces X_{0} ∩ X′_{1} and X′_{0} ∩ X_{1}, which must be empty. Collarable maps, as here defined, do not form a subcategory of Top: the composition of two 1-collarable maps is 2-collarable, in a graded category (see 2.9).

A 1-collared map is a map equipped with a collar. A collared map is a sum f = f_{0} + f_{1} of topological maps, where f_{0} is a homeomorphism (considered as 0-collared) and f_{1} is 1-collared.

2.2. Collarable cospans. Similarly, we say that a topological cospan

u = (u^{−}: X^{−} → X^{0} ← X^{+} : u^{+}),  (17)

(i) is 0-collarable, or trivially collarable, if it is a pair of homeomorphisms, (ii) is 1-collarable, or admits a collar cospan, (or, simply, a collar), if there exists a cospan (U^{−}, U^{+}) formed of a pair of collars of its maps having disjoint images.

In other words, in the second case, we have two disjoint closed embeddings

U = (U^{−}: IX^{−} → X^{0} ← IX^{+} : U^{+}),  u^{α} = U^{α}(−, 0): X^{α} → X^{0},  (18)

where U^{α}(X^{α} ×[0, 1[) is open in X^{0}. (Examples have been recalled at the beginning of 2.1.)

Furthermore, we say that the cospan u is collarable if it admits a collarable decomposition, i.e. can be decomposed into a binary sum (1.3):

\begin{matrix}u = u_{0} + u_{1} = (X_{0}^{−} + X_{1}^{−} → X_{0}^{0} + X_{1}^{0} ← X_{0}^{+} + X_{1}^{+}), \\ u_{0} = (u_{0}^{−}: X_{0}^{−} → X_{0}^{0} ← X_{0}^{+} : u_{0}^{+}),  u_{1} = (u_{1}^{−}: X_{1}^{−} → X_{1}^{0} ← X_{1}^{+} : u_{1}^{+}), \end{matrix} (19)

where u_{0} is 0-collarable and u_{1} is 1-collarable (with collar cospan (U^{−}, U^{+})).

Again, the only cospan which is both 0- and 1-collarable is the empty cospan e_{1}(∅), and the collarable decomposition of a cospan is uniquely determined.

Cubical faces and degeneracy are inherited from ℂosp(Top)

∂_{1}^{α}u = X^{α},  e_{1}(X) = (id: X → X ← X : id)  (α = ±1).  (20)

We prove below (2.4) that collarable cospans are closed under the concatenation u+_{1} v of cospans. Topological spaces and collarable cospans form thus a transversally full weak double subcategory

ℂblc(Top) ⊂ ℂosp(Top).  (21)

Also here, we will write u = (u^{−}, u^{+}): X^{−} · → X^{+} to specify the cubical faces of u. Notice that a topological map, even if collarable, cannot be viewed as a collarable cospan, in general. The category of topological spaces is transversally embedded in ℂblc(Top), sending a map f: X → Y to the same transversal 0-map.

Collared cospans, where the 1-collared component is equipped with assigned collars, will be studied in Part III.

2.3. Studying concatenation. Let us be given a concatenation w = u +_{1} v of two collarable cospans, with the following collarable decompositions (2.2):

u = (u^{−}, u^{+}): X^{−} · → X^{+},  u = u_{0} + u_{1},  ((u_{1}^{−}, u_{1}^{+}) has collar (U^{−}, U^{+})),
v = (v^{−}, v^{+}): Y^{−} · → Y^{+},  v = v_{0} + v_{1},  (v_{1}^{−}, v_{1}^{+}) has collar (V^{−}, V^{+}),

X^{+} = A = Y^{−}.

(22)

In order to prove that w is collarable, we remark that the middle space A inherits a decomposition into four clopens (1.3)

A_{ij} = X_{i}^{+} ∩ Y_{j}^{−},  A = A_{00} + (A_{10} + A_{01} + A_{11})  (i, j = 0, 1),  (23)

which we reorganise as above into two, A_{00} (on which both cospans are trivially collarable) and its complement.

This decomposes the trivially collarable cospan u_{0} into a sum u′ + u′′ of two trivially collarable cospans with ∂_{1}^{+}u′ = A_{00}; analogously, v_{0} = v′ + v′′ with ∂_{1}^{−}v′ = A_{00}. The decompositions u = u′+(u′′+u_{1}) and v = v′+(v′′+v_{1}) give a decomposition of w = u+_{1}v as

u +_{1} v = (u′ +_{1} v′) + ((u′′ + u_{1}) +_{1} (v′′ + v_{1})),  (24)

where the first component w_{0} = u′ +_{1} v′ is 0-collarable. We prove below (Thm. 2.4) that the second is 1-collarable.

The argument can be better understood looking at the following example, where the spaces X^{−}, A and Y^{+} are discrete (with four points, each), and each A_{ij} is a singleton

X^{−} → X^{0} ← A → Y^{0} ← Y^{+}  (25)

•  ^{•}  ^{•}  ^{•}  ^{•}  (A_{00})^{•}  ^{•}  ^{•}  ^{•}  (A_{01})^{•}  ^{•}  ^{•}  ^{•}  (A_{10})^{•}  ^{•}  ^{•}  (A_{11})

The bottom row contains the component w_{0} of w, constructed by concatenating the restrictions of u_{0} and v_{0} to A_{00}. All the rest yields the component w_{1}, whose ‘left’ and ‘right’ collars are suggested by thick segments; notice that the collar of w_{1}^{−} is produced by the collars of u_{1}^{−} and part of the ones of v_{1}^{−}; on the other hand, the two dashed collars, which are pasted at A_{11} in the pushout, do not intervene in the collars of the composite. (One can get a similar example for cobordisms between 1-manifolds, replacing the singletons with circles and the 1-dimensional spaces with spheres with holes).

We speak of a 1-collarable concatenation when X_{0}^{+} ∩ Y_{0}^{−} = ∅, so that the trivial part is empty and u +_{1} v admits a collar; in this case, concatenation is computed by a pushout which will be said to be 1-collarable - and will be proved to be homeomorphic to a standard homotopy pushout (Thm. 2.5).

Thus, a concatenation of collarable cospans decomposes into the sum of two concatenations: the first is computed by a trivial pushout of homeomorphisms (preserved by any functor), and the second by a 1-collarable pushout, which will be proved to be respected - in suitable ways - by most (co)homotopy and (co)homology functors.

2.4. Theorem. [Concatenating collarable cospans] Collarable cospans of topological spaces are stable under concatenation in Cosp(Top). As a consequence, the same holds for their transversal maps.

More precisely, a concatenation w = u +_{1} v of collarable cospans has a collarable decomposition as in (24); in particular, w is 1-collarable if and only if the clopen X_{0}^{+}∩Y_{0}^{−}, on which u and v are both 0-collarable, is empty.

Proof. We have to prove that the second component of the last term in (24) admits a collar. To simplify notation, we will assume that A_{00} = X_{0}^{+} ∩ Y_{0}^{−} is empty (in figure (25), this amounts to discard the bottom row) and prove that then the concatenation w = u +_{1} v admits a collar. Note that, in A, we have now: X_{0}^{+} ⊂ Y_{1}^{−} and Y_{0}^{−} ⊂ X_{1}^{+}.

The cospan w = (w^{−}, w^{+}) is computed by a (distinguished) pushout in Top, over the common face A = X^{+} = Y^{−}

[figure: Z^{0}; X^{0}; j^{−}; Y^{0}; j^{+}; w^{−} = j^{−}u^{−}, w^{+} = j^{+}v^{+}.; X^{−}; u^{−}; A; u^{+}; v^{−}; Y^{+}; v^{+}]  (26)

The maps u^{+} and v^{−} are closed embeddings, and the set Z^{0} (the pasting of X^{0} and Y^{0} over A) is determined as

Z^{0} = j^{−}(X^{0}) ∪ j^{+}(Y^{0}),  j^{−}(X^{0}) ∩ j^{+}(Y^{0}) = j^{−}u^{+}(A) = j^{+}v^{−}(A).  (27)

Therefore, the mappings j^{α} are injective and Z^{0} has the finest topology which makes them continuous. Moreover, both j^{α} are closed embeddings: if C is closed in X^{0}, then the set j^{−}(C) has pre-image C in X^{0} and v^{−}((u^{+})^{−1}(C)) in Y^{0}; since both are closed, so is j^{−}(C) in Z^{0}.

We construct now a collar cospan W = (W^{−}, W^{+}) for w. Following again the guideline of figure (25), the extension W^{−}: IX^{−} → Z^{0} of w^{−} is defined separately on the components of IX^{−} = IX_{1}^{−} + IX_{0}^{−}, using the collars U^{−} and V^{−}, respectively, and is a closed embedding

[figure: Z^{0}; IX_{1}^{−}; U^{−}; X^{0}; j^{−}; Y^{0}; j^{+}; IY_{1}^{+}; V^{+}; w^{−} = j^{−}u^{−},; w^{+} = j^{+}v^{+},; X_{1}^{−}; d^{−}; X^{−}; u^{−}; X_{0}^{0}; A; u^{+}; v^{−}; IY_{1}^{−}; V^{−}; a = u_{0}^{−},; b = u_{0}^{+},; X_{0}^{−}; a; X_{0}^{+}; ⊂; b; IY_{1}^{−}; V^{−}]  (28)

\begin{matrix}W_{1}^{−} = j^{−}U^{−}: IX_{1}^{−} → Z^{0},  W_{1}^{−}(x, 0) = j^{−}U^{−}(x, 0) = j^{−}u^{−}(x) = w^{−}(x), \\ W_{0}^{−} = j^{+}.V^{−}.Ib^{−1}.Ia: IX_{0}^{+} → IX_{0}^{0} → IX_{0}^{+} ⊂ IY_{1}^{−} → Y^{0} → Z^{0}, \\ W_{0}^{−}(x, 0) = j^{+}.v^{−}.b^{−1}a(x) = j^{−}.u^{+}.b^{−1}a(x) = j^{−}u^{−}(x) = w^{−}(x). \end{matrix}  (29)

The extension W^{+} is defined in the symmetric way, using V^{+} on IY_{1}^{+} and U^{+} on IY_{0}^{+}. The images of W^{α} are disjoint, because:

Im(W^{−}) = Im(j^{−}U^{−}) ∪ Im(j^{+}V^{−}), Im(W^{+}) = Im(j^{+}V^{+}) ∪ Im(j^{−}U^{+}), Im(j^{−}U^{−}) ∩ Im(j^{+}V^{+}) ⊂ Im(j^{−}U^{−}) ∩ (Im(j^{−}) ∩ Im(j^{+}))

= Im(j^{−}U^{−}) ∩ Im(j^{−}u^{+}) = ∅,

Im(j^{−}U^{−}) ∩ Im(j^{−}U^{+}) = j^{−}(ImU^{−} ∩ ImU^{+}) = ∅,

(30)

and symmetrically.

Finally W^{−}(X^{−}×[0, 1[) is open in Z^{0}, since its pre-images in X^{0} and Y^{0} are, respectively

U^{−}(X_{1}^{−}×[0, 1[) ∪ b(X_{0}^{+}×[0, 1[),  ∅ ∪ V^{−}(X_{0}^{+}×[0, 1[).  (31)

2.5. Theorem. [Collarable pushouts and homotopy] (a) A pushout in Top of two collarable maps f and g is a homotopy pushout and satisfies the weak universal property 1.5(a), only concerning existence.

(b) More particularly, a 1-collarable pushout (2.3), i.e. the pushout of two collarable maps f and g whose clopens A′_{0}, A′′_{0} (where both f and g restrict to homeomorphisms) do not meet, is homeomorphic to the standard homotopy pushout.

In fact, the canonical map p from the standard homotopy pushout (P; u, v; λ) to the ordinary pushout (B, u′′, v′′), which collapses the cylinder IA, is a homotopy equivalence, with a quasi inverse h: B → P, which is a homeomorphism

[figure: X; u; u′′; u′; λ; A; f; g; P; p; B; h; P; Y; v; v′′; v′]  (32)

(c) The same, as in (b), holds for the pushout of a closed embedding f along a 1-collarable map g (and symmetrically). Note. This statement has some similarity with a classical fact, stating that the ordinary pushout of any map along a cofibration is a homotopy pushout.

Proof. Point (a) is a consequence of (b), since a pushout of homeomorphisms is trivially a homotopy pushout. We prove (c), the proof of (b) being similar. The map g has a collar G: IA → Y.

The standard homotopy pushout (P; u, v; λ) is constructed as in (12), with u, v closed embeddings

P = (X + IA + Y)/∼,  [f(a)] = [a, 0],  [g(a)] = [a, 1] (a ∈ A).  (33)

We define u′: X → P as the obvious composite

u′ = (X ⊂ X + IA + Y → (X + IA + Y)/∼).  (34)

Now, Y has a closed cover (Y_{1}, Y_{2}, Y_{3}) with two possibly non-empty intersections

\begin{matrix}Y_{1} = G(A×[0, 1/2]),  Y_{2} = G(A×[1/2, 1]),  Y_{3} = (Y \ G(A×[0, 1[), \\ Y_{1} ∩ Y_{2} = F (A×\{1/2\}),  Y_{2} ∩ Y_{3} = G(A×\{1\}). \end{matrix}  (35)

We define v′: Y → P as the pasting of the following closed embeddings

\begin{matrix}v_{1} : Y_{1} → P, & v_{1}G(a, t) = [a, 2t] & (0 ⩽ t ⩽ 1/2), \\ v_{2} : Y_{2} → P, & v_{2}G(a, t) = [G(a, 2t − 1) & (1/2 ⩽ t ⩽ 1), \\ v_{3} : Y_{3} → P, & v_{3}(y) = [y], \end{matrix}  (36)

which agree on the intersections, because

v_{1}G(a, 1/2) = [a, 1] = [g(a)] = [G(a, 0)],  v_{2}G(a, 1) = [G(a, 1)] = v_{3}(G(a, 1)). (37)

Now, u′ and v′ agree on A

u′(f(a)) = [f(a)] = [a, 0] = v′G(a, 0) = v′(g(a),  (38)

and define a closed embedding h: B → P which forms commutative triangles in (32). It is surjective, hence a homeomorphism. Finally, it is easy to prove that p (which is not injective) and h are inverse up to homotopy.

2.6. Cobordisms. We will use the weak double subcategory ℂob(n) ⊂ ℂblc(Top) of n-dimensional manifolds and their cobordisms. Explicitly:

(a) a 0-cube is any n-dimensional compact manifold (without boundary);

(b) a 1-cube, called a (generalised) cobordism, is a cospan between 0-cubes which can be decomposed as u = u_{0} + u_{1} in a trivial part u_{0}, consisting of a pair of homeomorphisms of compact n-manifolds, together with a cobordism part u_{1} = (u_{1}^{−}: X_{1}^{−} → X_{1}^{0} ← X_{1}^{+} : u_{1}^{+}), where X_{1}^{0} is a compact (n+1)-manifold with boundary and the maps u_{1}^{α}: X_{1}^{α} → X_{1}^{0} are disjoint closed embeddings whose images cover the boundary of X_{1}^{0}; then, the existence of the collar-pair (U^{−}, U^{+}) is a well-known consequence (e.g., see [22], 6.2 and [4]);

(c) transversal n-maps are all the natural transformations (in Top) between such n-cubes (n = 0, 1); notice that 1-maps are double cells.

One has only to check that, in the 1-collarable pushout (26) for two consecutive cobordisms, the central space Z^{0} is again an (n + 1)-manifold, with boundary covered by the new collars W^{α}. But all this is well known.

2.7. Pairs of spaces. We will also use the category Top_{2}, of relative pairs (X, A) of topological spaces, in the usual sense of Algebraic Topology: A is a subspace of X; a map f: (X, A) → (Y, B) is a continuous mapping X → A which takes A into B.

This category gives rise to a weak double category ℂosp(Top_{2}) of relative cospans and to a weak double subcategory ℂblc(Top_{2}) of relative collarable cospans.

For the latter, we give the following two definitions. A collarable map of relative pairs f: (X, A) → (Y, B) is a relative map which has a collarable decomposition

f = f_{0} + f_{1} : (X_{0}, A_{0}) + (X_{1}, A_{1}) → (Y_{0}, B_{0}) + (Y_{1}, B_{1}),  (39)

into an isomorphism f_{0} of Top_{2} and a map f_{1} which admits a (relative) collar, i.e. a closed embedding F: X_{1}×I → Y_{1} such that:

f_{1} = F (−, 0): X_{1} → Y_{1},  F (X_{1}×[0, 1[) is open in Y,
F (A_{1}×I) is closed in B_{1},  F (A_{1}×[0, 1[) is open in B_{1}.

(40)

A collarable cospan of relative pairs is a cospan in Top_{2}

u = (u^{−}: (X^{−}, A^{−}) → (X^{0}, A^{0}) ← (X^{+}, A^{+}) : u^{+}),  (41)

which has a collarable decomposition, in the obvious sense extending the absolute case, 2.2.

2.8. Weak equivalences and homotopy invariance. Recall that every space X has a degenerate cospan e_{1}(X) and a cylindrical cospan E_{1}(X) (13), which are 0- and 1-collarable, respectively. We want to make them ‘equivalent’, also because, within cobordisms of n-manifolds, the cylindrical cospan is generally used as a weak identity, instead of the degenerate one. (Notice that this latter one is not ‘represented’ by an (n+1)-manifold with boundary.)

We say that a transversal 1-map f = (f^{−}, f^{0}, f^{+}): X → Y between topological cospans is a weak equivalence if it is special (i.e., f^{−} and f^{+} are identities) and its central component f^{0} is a homotopy equivalence. Thus, the canonical map p: E_{1}(X) → e_{1}(X)

(14) is a weak equivalence. We have already seen that it cannot be considered as a ‘homotopy equivalence’ in ℂosp(Top), because there are no arrows backwards, which is why this notion is too restrictive (and we do not even define it).

We say that two cospans X, Y with the same cubical faces are weakly equivalent if there exists a finite sequence of weak equivalences connecting them: X → X_{1} ← X_{2} → ... → X_{n} = Y.

Plainly, weak equivalence is closed under sum of cospans. Thus, every collarable cospan is weakly equivalent to a cospan which admits a collar.

A weak double functor F: ℂosp(Top) → 𝔸, with values in an arbitrary weak double category, will be said to be homotopy invariant if

(i) it identifies parallel transversal 0-maps X → Y (between topological spaces) which are homotopic; (ii) it sends weak equivalences f: X → Y between topological cospans to invertible (special) cells of 𝔸, and therefore weakly equivalent cospans to isomorphic 1-cubes of 𝔸. (This means isomorphic cospans when 𝔸 is a weak double category of cospans, and equal relations when 𝔸 is a strict double category of relations.)

We use the same terminology for weak double functors defined on substructures of ℂosp(Top), like the previously considered ℂblc(Top) and ℂob(n) (2.2, 2.6). Weak equivalences and homotopy invariance for cospans of relative pairs are similarly defined.

2.9. Higher collars. Extending the definitions of 2.1, one can form a graded category Col_{∗}(Top) of topological spaces and n-collarable maps, which will not be used in this Part.

After degree 0 (which means a homeomorphism), we say that a continuous mapping f: X → Y admits an n-collar if there is an extension to the n-cylinder I^{n}X = X×[0, 1]^{n}, which is a closed embedding

F: I^{n}X → Y,  f = F (−, 0, ..., 0): X → Y,  (42)

so that the subset F (X ×[0, 1[^{n}) is open in Y.

It is easy to verify that the composite gf with an m-collarable map g: Y → Z is collarable, of degree m + n. Indeed, leaving out the trivial cases, if G: I^{m}Y → Z is an m-collar of g, then gf can be extended to

G.I^{m}F: I^{m+n}X → I^{m}X → Y,  (43)

and G.(I^{m}F (X ×[0, 1[^{m+n}) = G.(F (X ×[0, 1[^{n})×[0, 1[^{m}) is open in Z.

3. Cohomotopy functors and the linearised TQFT

The circle S^{1} is a Frobenius object in the monoidal category of (isomorphism classes of) collarable cospans. Furthermore, Borsuk’s cohomotopy functors induce functors from collarable cospans to spans of sets. Restricting to manifolds and their cobordisms, we get spans of finite sets, and - by linearisation - topological quantum field theories. It would be interesting to characterise the TQFTs which can be obtained in this way.

3.1. The Frobenius structure on the circle. Let us consider the circle S^{1} in the weak double category ℂob(1) ⊂ ℂblc(Top) of 1-dimensional manifolds and their cobordisms (2.6), or rather in the associated involutive category Cob_{iso}(1) ⊂ Cblc_{iso}(Top) of isomorphism-classes of such cobordisms (cf. 1.1). These categories are equipped with the monoidal structure induced by the sum in the previous double structures.

S^{1} is an involutive Frobenius object in Cob_{iso}(1), in the sense of [16], 3.6, when equipped with the following cospans

η: ∅ · → S^{1},  ε = η^{♯}: S^{1} · → ∅,  (cups),
μ: S^{1} + S^{1} · → S^{1},  δ = μ^{♯}: S^{1} · → S^{1} + S^{1}  (trousers),

(44)

[figure: S^{1}; ∅; ∅; S^{1}; S^{1}; S^{1} + S^{1}; S^{1} + S^{1}; S^{1}; •]

The axioms are satisfied (strictly), in Cob_{iso}(1) and Cblc_{iso}(Top):

\begin{matrix}μ(η + 1) = 1 = μ(1 + η), & (ε + 1)δ = 1 = (1 + ε)δ, \\ (1 + μ)(δ + 1) = δμ = (μ + 1)(1 + δ). \end{matrix}  (45)

3.2. Cohomotopy functors. Every topological space S defines a Borsuk cohomotopy functor, which is homotopy invariant (in the classical sense)

π^{S} = [−, S]: Top → Set^{op},  π^{S}(X) = [X, S].  (46)

In particular, the spheres give the functors π^{n} = [−, S^{n}]. Restricting to suitable spaces X (e.g., CW spaces of dimension up to 2n − 1), the set π^{n}(X) has a canonical abelian group structure [3, 21, 14]. More generally, an arbitrary Moore space K′(A, n) is also used as the classifying space S, to give π^{n}(X; A), the n-th cohomotopy group with coefficients in the abelian group A (always under restrictions on X) [20].

The simplest non trivial example is π^{0}(X) = [X, S^{0}], which can be identified with the set of clopen subsets of X; then, π^{0}(f) = f^{∗} acts as counterimage.

We will also consider the following functors, for n ⩾ 2

[−, P_{n}]: Top → Set^{op},  [−, P^{2}] = [−, P_{2}]: Top → Set^{op},  (47)

where P_{2} = P^{2} is the real projective plane, and more generally P_{n} (called a pseudoprojective plane) is the quotient of the disc B^{2} modulo the obvious action of Z_{n} on the boundary (identifying each n-tuple of vertices of a regular n-gon inscribed in the boundary).

3.3. Theorem. [Collarable cospans and cohomotopy] Let us be given a pushout of collarable maps

\xymatrix{A \ar[r]^{g} \ar[d]_{f} & Y \ar[d]^{v} \\ X \ar[r]_{u} & B}  (48)

(a) Every cohomotopy functor π^{S} = [−, S]: Top → Set^{op} turns it into a weak pullback of sets, and extends therefore to a strict double functor ℂblc(Top) → ℝel(Set^{op}), which is homotopy invariant (as defined in 2.8). We will write it as

π^{S} = [−, S]: ℂblc(Top) → ℝel(Set)  (contravariant on the strict maps). (49)

In particular, this holds for π^{n} = [−, |S^{n}].

(b) In the same way, we obtain a homotopy invariant colax double functor with values in spans

π^{S} = [−, S]: ℂblc(Top) → 𝕊p(Set)  (contravariant on the strict maps). (50)

(c) If the classifying space S is discrete (as happens for π_{0} = [−, S^{0}]), every pushout (1) is transformed into a pullback of sets, which yields a homotopy invariant pseudo double functor

π^{S} = [−, S]: ℂblc(Top) → 𝕊p(Set)  (contravariant on the strict maps). (51)

Moving to the associated categories Cblc_{iso}(Top) and Sp_{iso}(Set) (1.1) and restricting to the full subcategory fcoTop of topological spaces with a finite number of clopens, then applying the linearisation functor defined in the Appendix (see 5.2), we get a functor with values in finitely generated modules

Cblc_{iso}(fcoTop) → Sp_{iso}(fSet) → fMod,  (52)

(which will be used below to construct a TQFT on manifolds). It is homotopy invariant, with respect to the weak-equivalence relation induced on the quotient Cblc_{iso}(fcoTop).

Proof. First, let us remark that, in (c), the pseudo double functor π^{S} is not strict: we cannot expect it to preserve the choices of pushouts and pullbacks we are using to define concatenation in these pseudo double categories. In (a), this fact does not appear because the composition of relations does not depend on the choice of pushouts or pullbacks we may use.

Point (a) is a straightforward consequence of Thms. 2.4, 2.5: a concatenation of collarable cospans decomposes into two: the first is computed by a trivial pushout, preserved by any functor, and the second by a 1-collarable pushout, which is a homotopy pushout. Now, it suffices to apply Thm 5.5, of the Appendix.

Point (b) is obvious. As to (c), let h, k: B → S be two maps whose homotopy classes coincide on X and Y. Since S is discrete, this means that hu = hv and ku = kv. Therefore, h = k.

3.4. An elementary example. Consider the involution-preserving functor

π^{0} = [−, S^{0}]: Cblc_{iso}(Top) → Sp_{iso}(Set);

it is monoidal, in the sense that it takes sums of spaces to product of sets.

Applying it to the involutive Frobenius object S^{1} (3.1), we get an involutive Frobenius object in the category Sp_{iso}(Set) of isomorphism-classes of spans. Writing 2 = {0, 1} the Boolean algebra of clopens of S^{1}, we have:

η: {0} · → {0, 1},  μ: {00, 01, 10, 11} · → {0, 1} (ε = η^{♯}, δ = μ^{♯}).  (53)

By elementary computation on cups and trousers, we see that ε and δ are mappings:

ε: 2 → 1,  δ: 2 → 2×2  (the Cartesian diagonal).  (54)

This Frobenius object has a peculiar property, which will obviously be preserved in its linearisation:

μδ = id: 2 · → 2.  (55)

Applying now the (monoidal) linearisation functor Sp_{iso}(fSet) → fVct on a field k (5.2), we get an ordinary Frobenius algebra on k^{2} (whose canonical basis we write e_{0}, e_{1})

η: k → k^{2},  η(a) = (a, a),
μ: k^{2} ⊗ k^{2} → k^{2},  μ((a, b) ⊗ (c, d)) = μ((ae_{0} + be_{1}) ⊗ (ce_{0} + de_{1}))
= μ(ace_{00} + ade_{11} + bce_{10} + bde_{11}) = (ac, bd),
ε: k^{2} → k,  ε(a, b) = a + b,
δ: k^{2} → k^{2} ⊗ k^{2},  δ(a, b) = ae_{00} + be_{11} = ae_{0} ⊗ e_{0} + be_{1} ⊗ e_{1}.

(56)

The (plainly non-degenerate) pairing is:

β = εμ: k^{2} ⊗ k^{2} → k,  β((a, b) ⊗ (c, d)) = ac + bd.  (57)

3.5. The Frobenius algebra associated to a pseudo-projective plane. Consider the colax double functor π^{∗} = [−, P^{2}]: ℂblc(Top) → 𝕊p(Set) (contravariant on the strict maps). It would be interesting to prove that it is actually a pseudo double functor, or at least that this holds true when restricting to ℂob(1).

In any way, applying π^{∗}, we do get a Frobenius object in Sp(Set), on the set (group) [S^{1}, P^{2}] = Z_{2}. Then, the cup is contractible and any map to P^{2} is nullhomotopic; the trousers T have the homotopy type of the space ‘eight’ T′ (two tangent circles) and the obvious quotient-map S^{1} + S^{1} → T′ yields an isomorphism π^{∗}(T) → Z_{2}×Z_{2} which will be used to identify these sets.

Therefore we get the following involutive Frobenius object in Sp_{iso}(Set)

η: {0} → Z_{2},  μ = sum: Z_{2}×Z_{2} → Z_{2} (ε = η^{♯}, δ = μ^{♯}).  (58)

When we linearise this Frobenius object on the field k, we get the vector space k^{2}, with Frobenius structure:

\begin{matrix}(a, b).(c, d) = (ac + bd, ad + bc), & δ(a, b) = a(e_{00} + e_{11}) + b(e_{01} + e_{10}), \\ η(a) = (a, 0), & ε(a, b) = a. \end{matrix}  (59)

The ‘handle operator’ μδ: S^{1} ·→ S^{1} and the torus εμδη: ∅ ·→ ∅ give the following linear maps

μδ: k^{2} → k^{2},  μδ(a, b) = 2(a, b),  εμδη: k → k,  εμδη(a) = 2a.  (60)

Notice that the multiplicative part is precisely the group-algebra k[Z_{2}]. In fact, the space ‘eight’ T′ is the sum of two pointed circles (S^{1}, x_{0}), and, in the cospan

μ = (S^{1} + S^{1} → T ← S^{1}),

the second map can be viewed as the usual ‘comultiplication’ of the H-cospace structure on the pointed circle. Now, the functor π^{∗} sends the first map to an isomorphism (which we have made into an identity), and the second to the multiplication of π^{∗}(S^{1}) = [S^{1}, P^{2}] = π_{1}(P^{2}).

More generally, the pseudo-projective plane P_{n} (3.2) gives a Frobenius algebra on the vector space k^{n}, whose multiplicative part is the group-algebra k[Z_{n}]

η: {0} → Z_{n},  μ = sum: Z_{n}×Z_{n} → Z_{n} (ε = η^{♯}, δ = μ^{♯}),  (61)

\begin{matrix}(a_{0}, ..., a_{n−1}).(b_{0}, ..., b_{n−1}) = (∑_{i+j=k} a_{i}b_{j})_{k=0,...n−1}, \\ δ(a_{0}, ..., a_{n−1}) = a_{0}(∑_{i+j=0} e_{ij}) + ... + a_{n−1}(∑_{i+j=n−1} e_{ij}), \\ η(a) = (a, 0, ..., 0),  ε(a_{0}, ..., a_{n−1}) = a_{0}. \end{matrix}  (62)

4. Extending homology and homotopy functors to cospans

Absolute and relative (co)homology functors can be extended to collarable cospans, using the Mayer-Vietoris sequence. Similar extensions hold for the fundamental-groupoid functor.

4.1. Theorem. [Collarable pushouts and homology] Let us be given a homology theory H_{∗}, defined on all pairs of topological spaces (and satisfying the axioms of Eilenberg-Steenrod).

(a) In a pushout (B; u, v) of collarable maps f, g (as in (48)), the maps u, v and uf = vg are closed embeddings, and we have a Mayer-Vietoris exact sequence

\xymatrix{... \ar[r] & H_{n}(A) \ar[r]^{(f_{∗},g_{∗})} & H_{n}(X) ⊕ H_{n}(Y) \ar[r]^{[u_{∗},−v_{∗}]} & H_{n}(B) \ar[r]^{d} & H_{n−1}(A) \ar[r] & ...} (63)

(b) More generally, a pushout ((B, B_{0}); u, v) of collarable relative maps

f: (A, A_{0}) → (X, X_{0}),  g: (A, A_{0}) → (Y, Y_{0}),  (64)

yields a Mayer-Vietoris exact sequence for relative homology

[figure: ...; H_{n}(A, A_{0}); (f_{∗},g_{∗}); H_{n}(X, X_{0}) ⊕ H_{n}(Y, Y_{0}); [u_{∗},−v_{∗}]; H_{n}(B, B_{0}); d; H_{n−1}(A, A_{0}); ...]  (65)

Proof. We prove (a), since the generalisation to the relative case is straightforward. By Thm. 2.5, we can assume that our pushout is 1-collarable, hence homeomorphic to a homotopy pushout, since the contribute of the trivial part plainly satisfies our thesis.

Let us replace the ordinary pushout with the standard homotopy pushout, in the usual construction B = (X +IA+Y)/ ∼. Writing u′, w, v′ the closed embeddings of X, IA and Y in B, the latter is covered by the following closed neighbourhoods X′, Y′ of u(X), v(Y)

\begin{matrix}X′ = u′(X) ∪ w(A×[0, 2/3]) ⊂ B, & Y′ = v′(Y) ∪ w(A×[1/3, 1]) ⊂ B, \\ A′ = X′ ∩ Y′ = w(A×[1/3, 2/3]). \end{matrix}  (66)

The embeddings A → A′, X → X′, Y → Y′ are homotopy equivalences, and induce isomorphisms in homology. It suffices therefore to apply the exact sequence of the triad (B, X′, Y′), provided we prove that the latter is a proper triad for H_{∗}, in the sense of Eilenberg-Steenrod ([7], p, 34).

In other words, we want to show that the following induced homomorphisms are isomorphisms

H_{n}(X′, A′) → H_{n}(B, Y′),  H_{n}(Y′, A′) → H_{n}(B, X′).  (67)

In fact, they are excision isomorphisms. Working on the first, we are exciding the subset Y′ \ A′, since B \ (Y′ \ A′) = X′. The topological hypotheses are satisfied:

(i) the subset Y′ \ A′ = v′(Y) ∪ w(A×]2/3, 1]) is open in B, (ii) its closure is contained in the interior of Y′:

cl(Y′ \ A′) = v′(Y) ∪ w(A × [2/3, 1]) ⊂ v′(Y) ∪ w(A×]1/3, 1]) = int(Y′). (68)

4.2. Extending homology theories. Let us be given a homology theory H_{∗} on Top_{2}. We already know, by Thm. 4.1 (and definition 5.3(b)) that H_{n} takes any pushout of collarable maps to an exact square of abelian groups. Therefore, by 5.5, every H_{n} has a canonical extension to a homotopy invariant (strict) double functor

H_{n}: ℂblc(Top) → ℝel(Ab),  (69)

with values in the strict double category of abelian groups and relations. Furthermore, if we restrict to the full substructure ℂblc(Top, H_{n−1}) ⊂ ℂblc(Top) determined by the topological spaces X whose reduced homology H̃_{n−1}(X) is trivial:

H̃_{n−1}(X) = Ker(H_{n−1}(X) → H_{n−1}{∗}),  (70)

we get a stronger result: the functor H_{n} takes a pushout of collarable maps to a pushout of abelian groups (see 4.3).

Thus, H_{n} has a unique involution-preserving extension to a homotopy invariant pseudo double functor

H_{n}: ℂblc(Top, H_{n−1}) → ℂosp(Ab).  (71)

Notice that the condition H̃_{n−1}(X) = 0 is only imposed on the cubical faces of cospans, not on their middle spaces. Of course, composing extension (71) with the projection ℂosp(Ab) → ℝel(Ab) we find again the previous extension of H_{n}; or, more precisely, a restriction of the latter.

4.3. Proposition. Let us be given a concatenation u +_{1} v of collarable cospans, in the usual notation u = (u^{−}: X^{−} → X^{0} ← X^{+} : u^{+}), v = (v^{−}: Y^{−} → Y^{0} ← Y^{+} : v^{+}) and X^{+} = A = Y^{−}.

(a) Each of the following conditions ensures that H_{n} preserves the concatenation u +_{1} v:

(i) the homomorphism u_{∗}^{+} = H_{n−1}(u^{+}): H_{n−1}(A) → H_{n−1}(X^{0}) is injective, (ii) the homomorphism v_{∗}^{−} = H_{n−1}(v^{−}): H_{n−1}(A) → H_{n−1}(Y^{0}) is injective.

More generally, it suffices to know that this pair of homomorphisms is jointly monic.

(b) These conditions are necessarily satisfied if H̃_{n−1}(A) = 0.

Proof. (a). In the composition w = u +_{1} v (8), the pushout gives rise to an exact sequence

H_{n}(A) → H_{n}(X^{0}) ⊕ H_{n}(Y^{0}) → H_{n}(Z^{0}) → H_{n−1}(A) → H_{n−1}(X^{0}) ⊕ H_{n}(Y^{0}). (72)

Saying that the pair (u_{∗}^{+}, v_{∗}^{−}) is jointly monic means that the last homomorphism is injective. Then the preceding one (the third) is null, and the second is surjective. Applying 5.3(d), we conclude that H_{n} preserves our pushout.

(b) For n > 1, our condition means H_{n−1}(A) = 0, which trivially implies (i) and (ii). For n = 1, it suffices to note that the terminal map t: A → {∗} can be factored as A → X^{0} → {∗}; therefore, if H_{n−1}(t) is injective, so must H_{n−1}(u^{+}) be (and, similarly, H_{n−1}(v^{−})).

4.4. Extending cohomology theories. Let us be given a cohomology theory H^{∗} on Top_{2}.

We could redo and adapt what we have done for homology theories, but duality can be used as a shortcut, since H^{∗} is a homology theory with values in the abelian category Ab^{op}. Moreover, ℝel(Ab) and ℝel(Ab^{op}) have the same weak structure and a dual strict structure. Finally, the notion of an exact square in an abelian category (5.3) is self-dual.

Thus, we conclude that H^{n} takes any pushout of collarable maps to an exact square of abelian groups, and has canonical extensions to homotopy invariant double functors, contravariant on the strict maps

H^{n}: ℂblc(Top) → ℝel(Ab),  H^{n}: ℂblc(Top_{2}) → ℝel(Ab).  (73)

4.5. Theorem. [Extending the fundamental-groupoid functor] Let us be given a pushout of collarable maps f, g, as in (48). Then the fundamental-groupoid functor Π_{1} : Top → Gpd turns it into a pushout of groupoids. Therefore, Π_{1} has a canonical extension to a pseudo double functor

Π_{1} : ℂblc(Top) → ℂosp(Gpd),  (74)

with values in the pseudo double category of small groupoids, their functors and their cospans.

Proof. As in the proof of 4.1, one replaces A, X, Y with homotopy equivalent spaces A′, X′, Y′, so that B is covered by the interior of X′, Y′. Then one applies R. Brown’s version of the Seifert-van Kampen theorem for the fundamental groupoid [5, 6].

4.6. Using absolute homology groups and relations. We end this section by a few elementary computations, to show how these invariants can work.

Consider the following collarable cospans S^{1} · → S^{1}

S^{1}  S^{1}  S^{1}  S^{1}  (75)

[figure: S^{1}; S^{1}; K; S^{1}; •; S^{1}]

The first is the cylinder E_{1}(S^{1}), while the second, K, results from ‘reversed’ embeddings of S^{1} into the cylinder (the coequaliser of such embeddings being the Klein bottle). The functor H_{1} sends these cospans to the following relations Z · → Z, respectively

id: Z → Z,  −: Z → Z,  ω: Z · → Z,  Ω: Z · → Z,  (76)

(ω is the least endorelation, which only associates 0 to 0, while Ω is the greatest).

More interesting examples can be obtained on non-connected 1-manifolds, like topological sums of circles S^{1} + ... + S^{1}, or, more generally, n-dimensional manifolds.

Working with Cosp(Ab), we get lax double functors H_{n}: ℂblc(Top) → ℂosp(Ab) which are pseudo (i.e., preserve concatenation up to isomorphism) under suitable hypotheses (4.2, 4.3). This version can be used to distinguish the collarable cospans above, in figure (75), from other similar ones, having some non connected component in the middle space of the cospan (which would be ignored by the homology relation).

Similar results hold for the fundamental-groupoid functor (see 4.5).

4.7. Using relative homology groups and relations. Tangles live in an ambient space: say the disc B^{2} for domain and codomain, and the solid cylinder B = B^{2}×[0, 1] for the tangles themselves.

Relative homology (in these ambient spaces) allows one to distinguish among the following generators, viewed as relative cospans (the tangle is always written as T)

(B^{2}, S^{0}) → (B, T) ← (B^{2}, S^{0}),  (B^{2}, S^{0}) → (B, T) ← (B^{2}, S^{0}),
(B^{2}, S^{0}) → (B, T) ← (B^{2}, ∅),  (B^{2}, ∅) → (B, T) ← (B^{2}, S^{0}), (77)

[figure: a; b; a; b; a; b; a; b; L^{+}; a; b; a; b; •; •; •; •; •; •; •; •; •; •; •; •; •]

In fact, easy computations show that they induce, via H_{1}, the following relations

id: Z → Z,  −: Z → Z,  0: Z → 0,  0^{♯}: 0 · → Z.  (78)

Note, however, that we cannot distinguish in this way the tangle L^{+} from L^{−} (which has the opposite choice of the arc which passes over). Thus, the invariants which we are obtaining are not sufficient to classify the isotopy classes of tangles.

On the other hand, again, relative-homology cospans (instead of relations) can detect components of the tangle which are not connected to domain and codomain, as in fig. 1 of the Introduction.

5. Appendix: Extending functors to cospans and relations

The abelian category of left R-modules, on a fixed (unitary) ring R is written Mod. In the preceding sections, we have used a field k, and written Vct the corresponding category (but rings and modules can also be used there).

5.1. Theorem. [Extending functors to cospans] (a) An arbitrary functor F: X → Y between categories with (a full choice of) pushouts has a canonical extension to a lax double functor

F: ℂosp(X) → ℂosp(Y),  (79)

which is a pseudo double functor if F preserves pushouts (and is strict if F preserves the distinguished choices).

(b) For X = Top, we have a lax extension ℂblc(Top) → ℂosp(Y), which is a pseudo double functor if F preserves pushouts of collarable maps.

(c) For the contravariant case, take Y = C^{op}, where C is a category with pullbacks. Now, we have a colax extension (contravariant on the strict maps)

F: ℂosp(X) → 𝕊p(C),  (80)

which is pseudo if F transforms pushouts of X into pullbacks of C.

Proof. We only have to prove (a). Let the extension F act on cospans and their transversal maps in the obvious way:

\begin{matrix}u = (u^{−}: X^{−} → X^{0} ← X^{+} : u^{+}), & F u = (F u^{−}: F X^{−} → F X^{0} ← F X^{+} : F u^{+}), \\ f = (f^{−}, f^{0}, f^{+}): u → v, & F f = (F f^{−}, F f^{0}, F f^{+}). \end{matrix} (81)

Degenerate cospans are strictly preserved, but a concatenation in X of consecutive cospans u, v (with notation as in (8)) gives rise to a comparison special transversal map F [u, v]

F [u, v]: F u +_{1} F v → F (u +_{1} v),  (82)
[figure: F Z^{0}; Z^{0}; •; X^{0}; Y^{0}; F X^{0}; F Y^{0}; X^{−}; A; Y^{+}; F X^{−}; F A; F Y^{+}]

Plainly, if F preserves pushouts (resp. distinguished pushouts), then these comparisons are invertible (resp. identities) and the extension is a pseudo (resp. strict) double functor.

5.2. Theorem. [Linearisation] The free R-module functor F: fSet → fMod, from finite sets to finitely generated R-modules, can be extended to a monoidal functor Sp_{iso}(fSet) → fMod, defined on the category of isomorphism classes of spans.

Also the latter is monoidal: it takes Cartesian product of sets to tensor product of (free) modules.

Proof. Let u = (f, g) = (X ← A → Y) be a span in fSet. Its linearised homomorphism is defined on the standard basis X of F X as

F u: F X → F Y,  (F u)(x) = ∑_{a∈A, f(a)=x} g(a).  (83)

One can notice that F u is represented by the matrix (n_{xy}), where n_{xy} is the number of points a ∈ A such that f(a) = x and g(b) = y. The reversed span gives thus the transpose matrix. (The extension could be said to be an involution-preserving functor with values in the category of free modules with an assigned basis.)

Indeed, given a consecutive span v = (h, k) = (Y ← B → Z), with composition u + v over the pullback C = {(a, b) | g(a) = h(b)}:

(F v)(F u)(x) = ∑_{a∈A, b∈B, f(a)=x, h(b)=g(a)} k(b) = ∑_{(a,b)∈C, f(a)=x} k(b).  (84)

Notice that a similar procedure for relations, instead of spans, would not be consistent with composition.

5.3. Additive relations and exact squares. We are also interested in extending functors to double categories of relations, like ℝel(Mod) or ℝel(Set). Let us begin by recalling the first case.

A relation a: A · → B between R-modules (also called an additive, or linear, relation) can be equivalently defined as:

(i) a submodule of the direct sum A ⊕ B, (i′) a quotient of the direct sum A ⊕ B, (ii) an equivalence class [v]: A · → B of spans having the same image in A ⊕ B (or the same pushout), (ii′) a class [u]: A · → B of cospans giving the same quotient of A ⊕ B (or having the same pullback), (iii) a jointly monic span A · → B, (iii′) a jointly epic cospan A · → B.

Relations A · → B, with a fixed domain and codomain, form an ordered set, by means of the inclusion of submodules of A ⊕ B, or by the existence of a special transversal map between representative (co)spans. The composition of (co)spans induces a (well known) composition of relations, which is strictly categorical. We have thus a category Rel(Mod), which is a quotient of the categories Sp_{iso}(Mod) and Cosp_{iso}(Mod).

Now, in order to extend functors to additive relations, we must know which squares in Mod

\xymatrix{A \ar[r]^{f} \ar[d]_{g} & B \ar[d]^{h} \\ C \ar[r]_{k} & D}  (85)

become bicommutative in Rel(Mod), which means that:

hf = kg,  gf^{♯} = k^{♯}h: B → C.  (86)

This has been determined in a 1966 paper, by Hilton [13]. An arbitrary square (85) of R-homomorphisms has an associated sequence

\xymatrix{0 \ar[r] & A \ar[r]^{(f,g)} & B ⊕ C \ar[r]^{[h,−k]} & D \ar[r] & 0}  (87)

The square is:

(a) commutative if and only if the sequence is of order two,

(b) exact (by definition) if and only if the sequence is exact in the central object,

(c) a pullback if and only if the sequence is exact in the central and left objects,

(d) a pushout if and only if the sequence is exact in the central and right objects. It is easy to see that the square (85) is exact if and only if:

(e) for all b ∈ B, c ∈ C with h(b) = k(c), there exists some a ∈ A such that f(a) = b, g(a) = c, if and only if the underlying square of mappings is a weak pullback in Set (i.e., it verifies there the existence part of the universal property).

(A self-dual diagrammatic characterisation of exact squares will be given below, in 5.6). It is now straightforward to verify that the square (85) becomes bicommutative in Rel(Mod) if and only if it is exact in Mod.

Of course, we also have a strict double category ℝel(Mod) (studied in [11]), which is a quotient of the weak double categories of spans and cospans: objects are modules, 1-cubes are relations, 0-maps are homomorphisms and 1-maps (or double cells) are ‘laxcommutative’ squares

\xymatrix{A \ar[rr]^{f} \ar[dd]|{•}_{a} &  & A′ \ar[dd]|{•}^{b} \\  & ⩽ &  & ga ⩽ bf. \\ B \ar[rr]_{g} &  & B′}  (88)

Notice that this double category is flat [11]: a double cell is determined by its boundary. All this extends to arbitrary abelian categories, except the characterisation of exact

squares as weak pullbacks, or by property (e), which is based on elements. It is still true that a square in an abelian category is exact if and only if it is bicommutative in the corresponding category of relations.

5.4. Relations of sets. For sets (or for a general regular category), these dual constructions of relations - as classes of spans or cospans - split into two distinct constructions.

Thus, a relation a: A · → B between sets amounts, equivalently, to:

(i) a subset of the Cartesian product A×B, (ii) an equivalence class [v]: A · → B of spans having the same image in A×B, (iii) a jointly monic span v: A · → B.

They are composed by pullback, and form a category Rel(Set), which is a quotient of Sp_{iso}(Set).

Again, we also have a larger structure, the (flat) double category ℝel(Set), which is a quotient of the weak double category 𝕊p(Set).

On the other hand, the dual construction, say Corel(Set) = Rel(Set^{op}), is not used here. (A corelation A · → B is a quotient of the sum A + B; they compose by pushouts. Corelations between sets, or finite sets, were studied in the 70’s under the name of transductors, for applications to the theory of devices, since they can be viewed as formalising ‘boxes’ of electrical connections between two sets of terminals. See [19] and its references.)

Also here, it is straightforward to see that a square (85) of mapping of sets becomes bicommutative in Rel(Set) if and only if it satisfies the condition 5.3(e), if and only if it is a weak pullback in Set. More elegant, self-dual characterisations are given below (5.6), but are not used in this article.

5.5. Theorem. [Extending functors to relations] Let Y be the category Mod of Rmodules (resp. Set). A functor F: X → Y defined on a category X with (a full choice of) pushouts has a canonical extension to a lax double functor

F: ℂosp(X) → ℝel(Y).  (89)

The latter is a strict double functor if F sends pushouts of X to exact squares of modules (resp. weak pullbacks of sets); in this case, the extension is uniquely determined.

For X = Top, we have a similar extension ℂblc(Top) → ℝel(Mod), which is a strict double functor if F sends pushouts of collarable maps to exact squares of modules (resp. weak pullbacks of sets).

Proof. The existence of the extension is proved as in 5.1, taking into account the characterisation of bicommutative squares recalled above, in 5.3 (resp. 5.4).

Its uniqueness depends on the fact that the reversed relation f^{♯} of a morphism f: A → B is determined as its adjoint morphism in the 2-categorical structure, by the inequalities 1 ⩽ f^{♯}f and ff^{♯} ⩽ 1. Now, writing u^{♯} = (1: Y → Y ← X : u) the reversed cospan of a map u: X → Y in X, it is easy to see that there are special transversal maps in ℂosp(X)

e_{1}(X) → u +_{1} u^{♯},  u^{♯} +_{1} u → e_{1}(Y).  (90)

Therefore a double functor (89) which extends the original F must send u^{♯} to the adjoint of F u, and is thus uniquely determined on all cospans.

5.6. Semicartesian squares. We end by recalling a self-dual notion, which characterises the squares which become bicommutative within relations of sets or modules (or also in any abelian category).

A commutative square (f, g; h, k) in an arbitrary category is said to be semicartesian [8] if:

(i) given a span (f′, g′) which commutes with (h, k) and a cospan (h′, k′) which commutes with (f, g), the outer diamond commutes: h′f′ = k′g′

[figure: B; h; h′; •; f′; g′; A; f; g; D; •; C; k; k′]  (91)

Plainly, any pullback and any pushout is semicartesian. When pullbacks and (or) pushouts exist, semicartesian squares can be characterised by various equivalent properties based on such (co)limits: e.g., the span (f, g) and the pullback of (h, k) commute with the same cospans, or have the same pushout.

In Set, a square is semicartesian if and only if it is a weak pullback. In an abelian category, a square is semicartesian if and only if it is exact in the sense of Hilton (5.3(b)).

References

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[3] K. Borsuk, Sur les groupes des classes de transformations continues, C.R. Acad. Sci. Paris 202 (1936), 1400-1403.

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[10] M. Grandis, Higher cospans and weak cubical categories (Cospans in Algebraic Topology, I), Theory Appl. Categ. 18 (2007), No. 12, 321-347.

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[11] M. Grandis and R. Paré, Limits in double categories, Cah. Topol. Géom. Différ. Catég. 40 (1999), 162-220.

[12] M. Grandis and R. Paré, Adjoint for double categories, Cah. Topol. Géom. Différ. Catég. 45 (2004), 193-240.

[13] P.J. Hilton, Correspondences and exact squares, in: Proc. Conf. on Categorical Algebra, La Jolla 1965, Springer, Berlin 1966, pp. 254-271.

[14] S. Hu, Homotopy theory, Academic Press, New York 1959.

[15] C. Kassel, M. Rosso and V. Turaev, Quantum groups and knot invariants. Panoramas et Synthèses, 5. Soc. Math. de France, Paris 1997.

[16] J. Kock, Frobenius algebras and 2D topological quantum field theories, Cambridge University Press, Cambridge 2004.

[17] J. Morton, A double bicategory of cobordisms with corners, Preprint. Available at arXiv: math.CT/0611930.

[18] M. Mather, Pull-backs in homotopy theory, Can. J. Math. 28 (1976), 225-263.

[19] F. Parodi, Simmetrizzazione di una categoria. II. Processi di simmetrizzazione. Rend. Sem. Mat. Univ. Padova 44 (1970), 223-262.

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[21] E. Spanier, Borsuk’s cohomotopy groups, Ann. of Math. 50 (1949), 203-245.

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Dipartimento di Matematica Università di Genova Via Dodecaneso 35 16146-Genova, Italy

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