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Stream: learning: questions

Topic: Clutching Data as a Functor?

Ruby Khondaker (she/her) (Jul 28 2026 at 10:53):

I've been learning some algebraic topology from Tom Dieck's book. He defines the notion of clutching data for both sets and topological spaces. I will reproduce an equivalent definition for sets here.

A clutching datum is a pair of families of sets $(S_i)_{i \in I}$ and $(S_i^j)_{i \in I, j \in I}$ with $S_i^j \subset S_i$ and $S_i^i = S_i$, together with a family of transition maps $g_{j \leftarrow i} : S_i^j \to S_j^i$ satisfying the following properties:
1) $g_{i \leftarrow i} = \text{id}_{S_i}$
2) $g_{j \leftarrow i}(S_i^j \cap S_i^k) \subset S_j^k$
3) On $S_i^j \cap S_i^k$ we have $g_{k \leftarrow i} = g_{k \leftarrow j} \circ g_{j \leftarrow i}$.

I've been trying to think about whether there's a way to naturally package up this information into a functor of some kind, given that "1" and "3" look very similar to identity and composition preservation. I think I have some partial progress - it involves considering the category of sets and partial functions as enriched in posets, where we say for $f , g : A \to B$ that $f \leq g \iff f(a) = g(a) \text{ whenever } f(a) \text{ is defined }$. This is really just a restriction of the partial order on homsets in $\mathbf{Rel}$.

In this case, a clutching datum looks to be some kind of enriched functor from the indiscrete category on $I$ to $\mathbf{Pfn}$.

Condition $1$ can be phrased as $\text{id}_{S_i} \leq g_{i \leftarrow i}$, and condition $3$ looks like it can be phrased as $g_{k \leftarrow j} \circ g_{j \leftarrow i} \leq g_{k \leftarrow i}$. However, a little more thought shows that the domains don't quite match up! The domain of the LHS is $S_i^j \cap g_{j \leftarrow i}^{-1}(S_j^k)$, whereas really we want to impose that this is equal to $S_i^j \cap S_i^k$.

In other words, it is not obvious to me how to encode condition "2" categorically. It essentially says "if $g_{j \leftarrow i}(x) \text{ and } g_{k \leftarrow i}(x)$ are defined, then $g_{k \leftarrow j}(g_{j \leftarrow i}(x))$ is defined", which doesn't quite match the composition in $\mathbf{Pfn}$.

Another way of seeing this is that, given only an enriched functor, it is not automatic that $g_{j \leftarrow i}(x)$ is defined implies $g_{i \leftarrow j}(g_{j \leftarrow i}(x))$ is defined. All we are guaranteed is that their composite, when defined, equals the identity - but for all we know, their composite could be defined nowhere! That would still be consistent with the functor laws.

I tried thinking about whether I could consider a category of partial bijections instead, but had no luck. So, to my question. I am (vaguely) aware that "gluing data" of the form above can be profitably viewed as a category, and there are descent statements to the effect of "the category of gluing data is equivalent to the category of glued-up objects". What I am asking is - can this category of gluing data be viewed as a functor category of some kind? Or do we have to impose the functor-like conditions "1" and "3" manually, so to speak?

James Deikun (Jul 28 2026 at 11:30):

It might make sense to express this with restriction categories.

Nathanael Arkor (Jul 28 2026 at 11:33):

This seems quite reminiscent of Marco Grandis's perspective on manifolds as enriched categories in Cohesive categories and manifolds, which is closely related to restriction categories as James points out.

Nathanael Arkor (Jul 28 2026 at 11:35):

(That is, the definition of "clutching datum" looks more like an enriched category to me, than an enriched functor. However, a lax functor from an indiscrete category to a bicategory $\mathcal W$ is precisely the data of a $\mathcal W$-enriched category on that set of objects, so the two perspectives are closely related.)

Ruby Khondaker (she/her) (Jul 28 2026 at 11:36):

Fascinating, I’ll check that out! It reminds me of a remark I saw here a while back, that it can be helpful to think of lax monoidal functors as their own kind of structure rather than merely a structure-preserving morphism.