A remark on colimits and the infinite Grassmannian that I didn't understand in class

Question

My friend was lecturing and he wrote down $\operatorname{colim}_n\operatorname{Gr}_k^n$ to be the infinite Grassmannian $\operatorname{Gr}_k^\infty$. Then my teacher said that the maps from $\operatorname{Gr}_k^n \to \operatorname{Gr}_k^{n+1}$ are not clear and he was asked to write down explicitly what the maps were.

First I think they have a simple description so I don't understand what the confusion was about in class:

Suppose you are given inclusions of $\mathbb{R} \xrightarrow{i_1} \mathbb{R}^2 \xrightarrow{i_2}\dots$ A point $x_n$ in $\operatorname{Gr}_k^n$ is an embedding $V_k \xrightarrow{x_n} \mathbb{R}^n$. This can be viewed as a $k$-plane in $\mathbb{R}^{n+1}$ as $i_n \circ x$.

Second, I don't understand what he wrote when he was asked this question. He fumbled for a second, someone mumbled "... if you add a dimension", and then he said thank you and then wrote $\operatorname{colim}_n\operatorname{Gr}_k^n \to \operatorname{colim}_{n+1}\operatorname{Gr}_{k+1}^{n+1}$ on the board, almost as if this was a description of something. Everyone in class seemed to be okay but me.

What on earth was my friend trying to get at?

Answer 1

"If you add a dimension." This was a good hint, unfortunately your friend did not use it in the intended way.

As you pointed out, the infinite Grassmannian $\operatorname{Gr}_k^{\infty}$ is the colimit of the collection $\{\operatorname{Gr}_k^n\}_{n=k}^{\infty}$ where the maps $\operatorname{Gr}_k^n \to \operatorname{Gr}_k^{n+1}$ are the ones you defined. Another way of describing these maps is $\operatorname{Gr}_k(\mathbb{R}^n) \to \operatorname{Gr}_k(\mathbb{R}^n\oplus\mathbb{R}) = \operatorname{Gr}_k(\mathbb{R}^{n+1})$, $P \mapsto P\oplus\{0\}$. That is, these maps "add a dimension" to the ambient space where the $k$-dimensional subspaces live. Under this map, the dimension of the subspaces doesn't change, but their codimension does.

Your friend said there is a map $\operatorname{colim}_n \operatorname{Gr}_k^n \to \operatorname{colim}_n \operatorname{Gr}^{n+1}_{k+1}$, that is $\operatorname{Gr}_k^{\infty} \to \operatorname{Gr}_{k+1}^{\infty}$. This is true, it is induced by the family of maps $\operatorname{Gr}_k(\mathbb{R}^n) \mapsto \operatorname{Gr}_{k+1}(\mathbb{R}\oplus \mathbb{R}^n) = \operatorname{Gr}_{k+1}(\mathbb{R}^{n+1})$, $P \mapsto \mathbb{R}\oplus P$. That is, these maps "add a dimension" to both the ambient space and the $k$-dimensional subspaces. Under this map, the dimension of the subspaces change, but their codimension doesn't.

These two collections of maps give the following commutative diagram: $\require{AMScd}$ \begin{CD} \operatorname{Gr}_k^n @>>> \operatorname{Gr}_k^{n+1}\\ @V V V @VV V \\ \operatorname{Gr}_{k+1}^{n+1} @>>> \operatorname{Gr}_{k+1}^{n+2}\\ \end{CD}

You and your teacher are referring to the horizontal arrows, while your friend is talking about the vertical arrows.