- Is there a "recognition principle" for projective spaces?
- What categories are there with projective spaces for objects?

**Background:** Although the title is a nod to What is a metric space?, this question is really built on top of my questions about Hilbert spaces (What is an intuitive view of adjoints? (version 2: functional analysis), Can one do without Riesz Representation?, Linearity of the inner product using the parallelogram law). The answers to these questions have set me pondering about Hilbert spaces. In short, I'm wondering whether or not it is possible to change the slogan of Hilbert spaces from "Hilbert spaces are great because they are self-dual" to "Hilbert spaces are great because you don't have to mention the dual". So I'm looking at ways to remove any explicit mention of the dual from the basic constructions in Hilbert space theory. My current favourite replacement is with statements about complementary subspaces.

This leads to projective spaces since they are the first non-trivial family of subspaces of some Hilbert space. The subspace analogue of Riesz representation says that there is an isomorphism between $\mathbb{P}H$ and $\operatorname{Gr}_{\infty -1}(H)$ (the space of codimension one subspaces). I want to be able to treat the latter space as if it were a projective space *in its own right* and say that this is an "isomorphism of projective spaces". However, I don't want to explicitly construct a vector space of which $\operatorname{Gr}_{\infty - 1}(H)$ is the projective space since that vector space would be $H^*$. So I want a recognition principle for projective spaces and a suitable notion of "isomorphism".

An example of what the recognition principle might be like would be an expression of the structure of a projective space as an algebraic theory, perhaps a many-sorted one. That would also make it obvious what the morphisms were.

Obviously there are many versions of what the morphisms could be. What I would like is for the obvious functor from Hilbert spaces to projective spaces to be full and faithful but not *by construction*. That is, I don't simply want to take the category of Hilbert spaces and replace each Hilbert space by it's projective space because then to work out the morphisms between two projective spaces I first need to choose representing Hilbert spaces and I don't want to do that.

To give this a connection to something a little deeper than just a different way of working with Hilbert spaces, recall that in twisted K-theory the starting place is a bundle of projective spaces that *cannot* be globally lifted to a bundle of Hilbert spaces. Nonetheless, some constructions related to a Hilbert space still work on such a bundle because they don't depend on the actual choice of Hilbert space so you can make a local choice and then prove it independent of that choice (for example, the space of Fredholm operators). Is there a way to go directly to that construction without making local choices?