Dagger and Compact-Closed Categories

A dagger category is a category equipped with an involutive, identity-on-objects, contravariant functor satisfying . This dagger is an abstract adjoint, so that unitary maps with and self-adjoint maps make sense. A compact-closed symmetric monoidal category gives every object a dual with a unit and counit satisfying the snake or yanking equations, so that wires may bend. A dagger-compact category carries both structures compatibly, since exchanges cups and caps. The prototype is with the Hermitian adjoint, the arena of categorical-quantum-mechanics and, more broadly, of the diagrammatic categories for physics and of the pregroup meaning maps in DisCoCat. is its boolean shadow.

References

• Max Kelly and Miguel L. Laplaza, “Coherence for compact closed categories”, Journal of Pure and Applied Algebra 19 (1980), pp. 193-213
• Peter Selinger, “Dagger compact closed categories and completely positive maps”, ENTCS 170 (2007), pp. 139-163
• Samson Abramsky and Bob Coecke, “A categorical semantics of quantum protocols”, LICS 2004, arXiv:quant-ph/0402130
• Bob Coecke, Mehrnoosh Sadrzadeh and Stephen Clark, “Mathematical Foundations for a Compositional Distributional Model of Meaning”, 2010, arXiv:1003.4394