## The Symplectic Sum: An Infuriating Construction

### Never Let Anybody Chase Your Diagrams for You

The symplectic sum construction upsets me. It is canonical, but not blessed by the categorical dialectic.

**Definition 1. ** A **s****ymplectic vector space **is a pair where is a vector space and is a non-degenerate anti-symmetric bilinear form.

**Example 2. **Consider Then,

is symplectic.

**Exercise 3. **Show that a symplectic vector space must have even dimension.

**Construction 4. **Let be symplectic vector spaces. Then,

makes into a symplectic vector space.

**Proof. **We show non-degeneracy. Fix Suppose that

for all Take and Then, But, we get

for all So, because is symplectic, we have Thus, for all By non-degeneracy of we have as well. This completes the proof.

Construction 4. is, in an intuitive sense, the canonical symplectic product form. Yet, *is not the categorical product. *

**Definition 5. **(The Symplectic Category) Let be symplectic vector spaces. A linear map is **symplectic **if A** symplectomorphism **is an invertible symplectic map.

**Exercise 6. **Check that the composition of symplectic maps is symplectic and that the identity is symplectic. So, there is a category **Symp **of symplectic spaces and symplectic linear maps.

**Exercise 7. **Show that every symplectic vector space is symplectomorphic to the one given as Example 2.

**Exercise 8. **Show that symplectomorphisms preserve volume. (Hint: Use non-degeneracy to show that is a volume form).

**“Counter”-example 9. **The projection is not symplectic.

**Proof. **Compute whenever . This completes the proof.