0th dimension: apoint

In homotopy theory, an n-truncated space is the analog of an n-dimensional space. Every n-manifold admits an n-truncated CW structure. In homotopy theory, a (-1)-space is either empty or a point, and a (-n)-space for n≥2 is a point. This convention works really really well, because the (-2)-homotopy of a space tells you if it exists, and the (-1)-homotopy tells you if it is inhabited. Then the 0-homotopy gives you the set of connected components, and the n-homotopy for n>0 is a group due to the cogroup structure on the n-sphere, and tells you about the loops in that dimension.

Of course, this is not the only way to rationalize (pun unintended) negative dimensions, but it remains consistent with higher category theory and the Dold-Kan correspondence.

>That is not what he means by complex dimensions. He is asking if the dimensionality of a space can be complex.

"Complex dimension" is a common term for the dimension of a complex manifold.

ex. A Riemann Surface is 1-(complex) dimensional. Hence the usual equivalence with complex algebraic curves.

Speak in terms someone in Calc 1 can understand

I thought fractals had non integer dimensions.

Let's start with CW complexes.

Start with a set of points, called the 0-skeleton of our space.

Now, we have a set of 1-cells, which are line segments. They are attached to the 1-cells via attaching maps: we have directions for which 0-cells each 1-cell is connected to. The only way we are allowed to attach 1-cells is by assigning their boundary points to various 0-cells.

A 2-cell is a 2-dimensional disk. But, like before, we can only attach 2-cells to our complex by attaching their boundaries to the 1-skeleton. The boundary of a 2-cell is a circle, so for each 2-cell we have an attaching map saying how its boundary fits into the 1-skeleton.

We can do this inductively as high as we want. The resulting space is our CW complex.

If we decided to stop at a certain dimension, so that our space has no (n+1) cells, then we call the space n-truncated.

An n-sphere is built by starting with one 0-cell, and then no k-cells until we get to k=n, in which case we map the entire boundary of a single n-cell onto the one point we started with.

Given a CW complex X, we can define something called the suspension of X. SX is defined to be the product X×I, of X with a 1-cell, under the quotient defined by the equivalence relations x×{0}=x'×{0} and x×{1}=x'×{1}. Intuitively, we take X, stretch it out to add an extra dimension, and the collapse the endpoints of this "cylinder" down into individual points.

The suspension of an n-sphere is homeomorphic to an (n+1)-sphere. As you might guess, the suspension of an n-truncated space is (n+1)-truncated. How can you prove this?

>yfw an algebraic stack can have negative dimension

Stacks can have negative dimension. A simpler example might be the empty set as the -1 dimensional sphere.
Fractals can have non-integer dimension, and complex dimensions are used in quantum field theory in dimensional regularisation. Physicists don't actually construct spaces with complex dimension, though - they just treat the dimension of spacetime as a complex parameter and take the d->4 limit.

>>In 0 dimensuons there is only a point so you are that point.

This got me thinking, if black holes have a sigularity of 0 dimension. Aren't all blackhole singularities the same point in space?

Just a lurking pleb here, so please correct my logic.

>negative dimensions
If the space is n-dimensional, then its dual is said to be (-n)-dimensional.

Think of it like an empty box. The box is empty, but that doesn't mean that it isn't a box.

No, just like a negative dimension doesn't really count as a dimension. Its a thought experiment

Anyone?

AHAHAHHAHAHA

Eleborate

Dimensionality refers to the intrinsic dimension, not the dimensionality of coordinates used to reference a relative position. It is non sequitir to suggest all zero-dimensional objects occupy the same position. All they have in common is that they all have no volume. Further, to say black holes "have" a singularity is speculation. The current mathematical models have a singularity. A singularity isn't a physical feature that has ever been observed. It is generally a sign that the current model begins to fail to be suitable near that point.

So if I understand correctly objects can have dimensions but do not exsist in dimensions?

And i knew it wasn't certain that there's a sigularity that's why i used the word 'if'. But thank you kind sir for expanding my limited knoweledge.

How is dimension defined for algebraic stacks, and can that definition be lifted to general stacks and/or ∞-stacks?

Yes. And as an aside, any statement made about elements in the empty set is true. For example, every element of the empty set is purple, or are 5 or are 10 and 0. This comes from some of our axioms of logic and specifically the logical operator p implies q. Something I always find interesting.

>Non integer dimension?

Negative dimensions? Fractional dimensions? Just sum all the positive integer dimensions and you will get a dimension of -1/12. Extend your logic from there.

Oh my god so triggered you huge faggot

>no mention of Hausdorff Dimension
OP, fractional dimensions exist, and have an intuitive meaning.

In 2-space, the measure of a square varies as the square of its side.

In 3-space, the measure of a cube varies as the cube of its side.

One can generalize this notion to come up with what fractional dimension means: if the measure of some region varies as the 1.5-th power of its "side", it is a 1.5-dimensional space.

en.wikipedia.org/wiki/Hausdorff_dimension

en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

>negative 3rd dimension
>evil counter parts

They don't exist you pop sci fag