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Carlo's answer is definitely pointing in the right direction: simplicial complexes or more generally, simplicial sets, are conjured up by most points points mentioned by the PO (certainly 1 3, 4. 5 perhaps, with a twist, and as for 2, no idea) .

Unfortunately, as indicated by Carlo's comments, it falls short on one requirement, that ants do not know anything about metric spaces (nor what you can build on them, such as differential geometry).

Poor ants live in a world whose departments of math contain only three courses:

1. finite combinatorics
2. topology (presumably also finite)
3. basic logic

Topological Data Analysis starts off with a cloud set of points immersed in a metric space (mostly euclidean $R^n$, but not necessarily).

Its main tool is persistent homology, which creates a filtration of simplicial complexes, thereby providing different views of $X$ at different resolution scales.

Where do these simplicial complexes come from ? They are Vietoris-Rips Complexes (see here; essentially you use the distance between groups of points to fill your simplexes).

So, no metric no Rips complex.

But (there is always a but in life): perhaps not all is lost.

What about creating a filtration of complexes by-passing entirely the metric?

Yes, sounds good, you may say, but how? Well, in ants world they have basic topology. So, for instance, suppose an ant goes from A to B, it can tell if during her trip she met point C (ie she can tell whether C is in some 'edge" between A and B). Similarly, given a set of distinguished points $A_0, \ldots A_n$ , she can tell whether they are indipendent, ie none of them lies in some slice of ant-world which is span by some subset. The independent subsets will become higher simplices (this approach is basically the one folks in matroid theory take)

Assuming this bare bone capability, Carlo's answer can indeed be vindicated: the ants build their filtration of complexes by selecting larger and larger finite subsets of their world.

Of course, unless their world is also finite, there is no guarantee that they will ever find out its final topology.

Carlo's answer is definitely pointing in the right direction: simplicial complexes or more generally, simplicial sets, are conjured up by most points points mentioned by the PO (certainly 1 3, 4. 5 perhaps, with a twist, and as for 2, no idea) .

Unfortunately, as indicated by Carlo's comments, it falls short on one requirement, that ants do not know anything about metric spaces (nor what you can build on them, such as differential geometry).

Poor ants live in a world whose departments of math contain only three courses (*):

1. finite combinatorics
2. topology (presumably also finite)
3. basic logic

Topological Data Analysis starts off with a cloud set of points immersed in a metric space (mostly euclidean $R^n$, but not necessarily).

Its main tool is persistent homology, which creates a filtration of simplicial complexes, thereby providing different views of $X$ at different resolution scales.

Where do these simplicial complexes come from ? They are Vietoris-Rips Complexes (see here; essentially you use the distance between groups of points to fill your simplexes).

So, no metric no Rips complex.

But (there is always a but in life): perhaps not all is lost.

What about creating a filtration of complexes by-passing entirely the metric?

Yes, sounds good, you may say, but how? Well, in ants world they have basic topology. So, for instance, suppose an ant goes from A to B, it can tell if during her trip she met point C (ie she can tell whether C is in some 'edge" between A and B). Similarly, given a set of distinguished points $A_0, \ldots A_n$ , she can tell whether they are indipendent, ie none of them lies in some slice of ant-world which is span by some subset. The independent subsets will become higher simplices (this approach is basically the one folks in matroid theory take)

Assuming this bare bone capability, Carlo's answer can indeed be vindicated: the ants build their filtration of complexes by selecting larger and larger finite subsets of their world.

Of course, unless their world is also finite, there is no guarantee that they will ever find out its final topology.

(*) on the funny side (apologies to serious MO fellows): trying to think of Ant's World I found out that is very much to my liking, especially the Departments of Math. A non Cantorian, non Dedekind paradise . Perhaps I should move there for a change:)

Carlo's answer is definitely pointing in the right direction: simplicial complexes or more generally, simplicial sets, are conjured up by all points mentioned by the PO.

Unfortunately, as indicated by Carlo's comments, it falls short on one requirement, that ants do not know anything about metric spaces (nor what you can build on them, such as differential geometry).

Poor ants live in a world whose departments of math contain only three courses:

1. finite combinatorics
2. topology (presumably also finite)
3. basic logic

Topological Data Analysis starts off with a cloud set of points immersed in a metric space (mostly euclidean $R^n$, but not necessarily).

Its main tool is persistent homology, which creates a filtration of simplicial complexes, thereby providing different views of $X$ at different resolution scales.

Where do these simplicial complexes come from ? They are Vietoris-Rips Complexes (see here; essentially you use the distance between groups of points to fill your simplexes).

So, no metric no Rips complex.

But (there is always a but in life): perhaps not all is lost.

What about creating a filtration of complexes by-passing entirely the metric?

Yes, sounds good, you may say, but how? Well, in ants world they have basic topology. So, for instance, suppose an ant goes from A to B, it can tell if during her trip she met point C (ie she can tell whether C is in some 'edge" between A and B). Similarly, given a set of distinguished points $A_0, \ldots A_n$ , she can tell whether they are indipendent, ie none of them lies in some slice of ant-world which is span by some subset. The independent subsets will become higher simplices (this approach is basically the one folks in matroid theory take)

Assuming this bare bone capability, Carlo's answer can indeed be vindicated: the ants build their filtration of complexes by selecting larger and larger finite subsets of their world.

Of course, unless their world is also finite, there is no guarantee that they will ever find out its final topology.

Carlo's answer is definitely pointing in the right direction: simplicial complexes or more generally, simplicial sets, are conjured up by most points points mentioned by the PO (certainly 1 3, 4. 5 perhaps, with a twist, and as for 2, no idea) .

Unfortunately, as indicated by Carlo's comments, it falls short on one requirement, that ants do not know anything about metric spaces (nor what you can build on them, such as differential geometry).

Poor ants live in a world whose departments of math contain only three courses:

1. finite combinatorics
2. topology (presumably also finite)
3. basic logic

Topological Data Analysis starts off with a cloud set of points immersed in a metric space (mostly euclidean $R^n$, but not necessarily).

Its main tool is persistent homology, which creates a filtration of simplicial complexes, thereby providing different views of $X$ at different resolution scales.

Where do these simplicial complexes come from ? They are Vietoris-Rips Complexes (see here; essentially you use the distance between groups of points to fill your simplexes).

So, no metric no Rips complex.

But (there is always a but in life): perhaps not all is lost.

What about creating a filtration of complexes by-passing entirely the metric?

Yes, sounds good, you may say, but how? Well, in ants world they have basic topology. So, for instance, suppose an ant goes from A to B, it can tell if during her trip she met point C (ie she can tell whether C is in some 'edge" between A and B). Similarly, given a set of distinguished points $A_0, \ldots A_n$ , she can tell whether they are indipendent, ie none of them lies in some slice of ant-world which is span by some subset. The independent subsets will become higher simplices (this approach is basically the one folks in matroid theory take)

Assuming this bare bone capability, Carlo's answer can indeed be vindicated: the ants build their filtration of complexes by selecting larger and larger finite subsets of their world.

Of course, unless their world is also finite, there is no guarantee that they will ever find out its final topology.

Carlo's answer is definitely pointing in the right direction: simplicial complexes or more generally, simplicial sets, are conjured up by all points mentioned by the PO.

Unfortunately, as indicated by Carlo's comments, it falls short on one requirement, that ants do not know anything about metric spaces (nor what you can build on them, such as differential geometry).

Poor ants live in a world whose departments of math contain only three courses:

1. finite combinatorics
2. topology (presumably also finite)
3. basic logic

Topological Data Analysis starts off with a cloud set of points immersed in a metric space (mostly euclidean $R^n$, but not necessarily.

Its main tool is persistent homology, which creates a filtration of simplicial complexes, thereby providing different views of $X$ at different resolution scales.

Where do these simplicial complexes come from ? They are Vietoris-Rips Complexes (see here; essentially you use the distance between groups of points to fill your simplexes).

So, no metric no Rips complex.

But (there is always a but in life): perhaps not all is lost.

What about creating a filtration of complexes by-passing the metric?

Yes, sounds good, you may say, but how? Well, in ants world they have basic topology. So, for instance, suppose an ant goes from A to B, it can tell if during her trip she met point C (ie she can tell whether C is in some 'edge" between A and B). Similarly, given a set of distinguished points $A_0, \ldots A_n$ , she can tell whether they are indipendent, ie none of them lies in some slice of ant-world which is span by some subset. The independet subsets will become higher simplices.

Assuming this bare bone capability, Carlo's answer can indeed be vindicated: the ants build their filtration of complexes by selecting larger and larger finite subsets of their world.

Of course, unless their world is also finite, there is no guarantee that they will ever find out its final topology.

Carlo's answer is definitely pointing in the right direction: simplicial complexes or more generally, simplicial sets, are conjured up by all points mentioned by the PO.

Unfortunately, as indicated by Carlo's comments, it falls short on one requirement, that ants do not know anything about metric spaces (nor what you can build on them, such as differential geometry).

Poor ants live in a world whose departments of math contain only three courses:

1. finite combinatorics
2. topology (presumably also finite)
3. basic logic

Topological Data Analysis starts off with a cloud set of points immersed in a metric space (mostly euclidean $R^n$, but not necessarily).

Its main tool is persistent homology, which creates a filtration of simplicial complexes, thereby providing different views of $X$ at different resolution scales.

Where do these simplicial complexes come from ? They are Vietoris-Rips Complexes (see here; essentially you use the distance between groups of points to fill your simplexes).

So, no metric no Rips complex.

But (there is always a but in life): perhaps not all is lost.

What about creating a filtration of complexes by-passing entirely the metric?

Yes, sounds good, you may say, but how? Well, in ants world they have basic topology. So, for instance, suppose an ant goes from A to B, it can tell if during her trip she met point C (ie she can tell whether C is in some 'edge" between A and B). Similarly, given a set of distinguished points $A_0, \ldots A_n$ , she can tell whether they are indipendent, ie none of them lies in some slice of ant-world which is span by some subset. The independent subsets will become higher simplices (this approach is basically the one folks in matroid theory take)

Assuming this bare bone capability, Carlo's answer can indeed be vindicated: the ants build their filtration of complexes by selecting larger and larger finite subsets of their world.

Of course, unless their world is also finite, there is no guarantee that they will ever find out its final topology.