Tammo tom Dieck's definition of Clutching

Question

Let $(U_j)_{j\in J}$ be a family of sets. Suppose for each $(i,j)\in J\times J$, a subset $U^j_i\subset U_i$ is given as well as a bijection $g^j_i:U^j_i\rightarrow U^i_j$. We call the families $(U_j, U^k_j,g^k_j)$ a clutching datum if

1. $U_j=U^j_j$ and $g^j_j=\text{id}$.
2. For each triple $(i,j,k)\in J\times J\times J$, the map $g^j_i$ induces a bijection $$g^j_i:U^j_i\cap U_i^k\rightarrow U^i_j\cap U^k_j$$ and $g^k_j\circ g^j_i=g^k_i$ holds, considered as maps from $U^j_i\cap U^k_i$ to $U^i_j\cap U^k_j$.

Given a clutching datum, we have an equivalence relation on the disjoint sum $\coprod_{j\in J}U_j$: $$x\in U_i\sim y\in U_j\iff x\in U_i^j\land g^j_i(x)=y.$$

Why is $\sim$ an equivalence relation on $\coprod U_j$? Let $z\in\coprod U_j$. There exists a unique $j\in J$ such that $z\in U_j\times\{j\}$. That is to say, there exists $x\in U_j$ such that $z=(x,j)$. Now $x\in U^j_j$ and $g^j_j(x)=x$. So $x\sim x$ but not $z\sim z$.

How can I rewrite the definition of the equivalence relation so that it doesn't make use of the identification $U_j\cong U_j\times\{j\}$? Is this the standard way of defining equivalence relations on disjoint unions?

Answer 1

tom Dieck is imprecise. He considers families of sets or topological spaces $(S_j \mid j \in J)$, but does not explain the meaning of $\coprod S_j$. This is of course the disjoint union of the sets $S_j$, but the only explicit mention occurs in a special case after Proposition 1.3.3:

Let $(X_j \mid j \in J )$ be a family of non-empty pairwise disjoint spaces. The set $\mathcal O = \{ U \subset \coprod X_j \mid U \cap X_j \text{ open for all } j \}$ is a topology on the disjoint union $\coprod X_j$. We call $(\coprod X_j, \mathcal O)$ the topological sum of the $X_j$.

By the way, there is no need to require that the $X_j$ are non-empty, but that is irrelevant. The main point is that he only considers pairwise disjoint spaces. In that case there is no need to use an undefined symbol $\coprod$, but one should simply take the ordinary union $\bigcup X_j$. Otherwise $U \cap X_j$ would not make any sense. See also Proposition 1.3.4 where tom Dieck states that the subspace topology of $X_j$ in $\coprod X_j$ is the original topology.

In my opinion he should have done this properly by introducing the disjoint union for an arbitrary family of $X_j$ as $\coprod X_j = \bigcup X_j \times \{j\}$. In that case the $X_j$ are no longer genuine subsets of $\coprod X_j$, we only have canonical injections $i_j : X_j \to \coprod X_j$. That this is urgently needed can be seen in section 1.3.8 where the colimit topology is introduced.

As he has done it, he blunders into a trap when he introduces the concept of clutching for a family of sets $(U_j \mid j \in J)$. To apply what he has introduced before, he must assume that the $U_j$ are disjoint. This is not explictly mentioned, but only under this assumption the equivalence relation $\sim$ is well-defined. Working with arbitary $U_j$ (which is certainly needed in many applications) requires $\coprod U_j = \bigcup U_j \times \{j\}$ and $$(x,i) \in U_i \times \{i\} \sim (y,j) \in U_j \times \{j\} \Leftrightarrow x \in U_i^j \text{ and } g_i^j(x) = y .$$

In my opinion tom Dieck makes a typical abuse of notation. Although $\coprod X_j \ne \bigcup X_j$, one identifies the $X_j$ with the subsets $X_j \times \{j\}$ of $\coprod X_j$ and writes $X_j = X_j \times \{j\}$. This is quite usual and does not cause real harm, but we should be aware that it is wrong in any formal aspect.