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Definition

A compact closed category, or simply a compact category, is a symmetric monoidal category in which every object is dualizable.

In particular, a compact closed category is a closed monoidal category, with the internal hom given by $[A,B]=A^{*}\otimes B$ (where $A^{*}$ is the dual of $A$).

More generally, if we drop the symmetry requirement, we obtain a rigid monoidal category, a.k.a. an autonomous category. Thus a compact category may also be called a rigid symmetric monoidal category or a symmetric autonomous category. A maximally clear, but rather verbose, term would be a symmetric monoidal category with duals for objects.

References

The Wikipedia entry isn’t bad:

• Wikipedia, Compact closed category .

Also see section 2.1 in:

• Peter Selinger, Dagger compact closed categories and completely positive maps. pdf