[Paper Review] Limits in dagger categories

Chris Heunen, Martti Karvonen|Edinburgh Research Explorer|Mar 18, 2018

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TL;DR

This paper introduces a new notion of limit in dagger categories that respects the dagger structure, ensuring uniqueness up to unitary isomorphism and compatibility with duality. It establishes that dagger limits commute with dagger colimits when the bifunctor is adjointable, generalizing known results and unifying special cases from quantum theory and category theory.

ABSTRACT

We develop a notion of limit for dagger categories, that we show is suitable in the following ways: it subsumes special cases known from the literature; dagger limits are unique up to unitary isomorphism; a wide class of dagger limits can be built from a small selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to a diagonal functor; dagger limits can be built from ordinary limits in the presence of polar decomposition; dagger limits commute with dagger colimits in many cases.

Motivation & Objective

• To define a notion of limit in dagger categories that respects the dagger structure and subsumes known special cases such as dagger products, equalizers, and kernels.
• To ensure that dagger limits are unique not just up to isomorphism but up to unitary isomorphism, reflecting the intrinsic symmetry of dagger categories.
• To establish conditions under which dagger limits commute with dagger colimits, generalizing classical results to the dagger setting.
• To show that a wide class of dagger limits can be constructed from a small set of basic limits, such as dagger equalizers, products, and intersections.
• To provide a dagger-adjoint formulation of limits via adjoint functors to the diagonal functor, extending classical category theory to the dagger context.

Proposed method

• Proposes a new definition of dagger limit that requires the limiting cone to be compatible with the dagger structure, ensuring that the limiting morphisms are preserved under duality.
• Uses the concept of unitary isomorphism to enforce uniqueness, replacing mere isomorphism in ordinary category theory to reflect the self-dual nature of dagger categories.
• Introduces the notion of adjointable bifunctors between index categories to ensure that limits and colimits commute in the dagger setting.
• Applies polar decomposition to construct dagger limits from ordinary limits in categories where such decomposition exists, such as in C*-categories.
• Establishes a dagger adjoint formulation of limits by showing that dagger limits arise as right adjoints to the diagonal functor in the dagger 2-category of dagger categories.
• Employs the 'way of the dagger' philosophy: all universal constructions must commute with the dagger functor, ensuring coherence between limits and colimits.

Experimental results

Research questions

• RQ1 How can the notion of limit in ordinary category theory be generalized to dagger categories in a way that respects the dagger structure?
• RQ2 What conditions ensure that dagger limits are unique up to unitary isomorphism rather than just isomorphism?
• RQ3 Under what conditions do dagger limits commute with dagger colimits in dagger categories?
• RQ4 Can a wide class of dagger limits be built from a small set of basic dagger limits, such as dagger equalizers and products?
• RQ5 To what extent can ordinary limits be transformed into dagger limits via polar decomposition or other structural properties?

Key findings

• Dagger limits are unique up to unitary isomorphism, which reflects the self-dual nature of dagger categories and ensures coherence under duality.
• A dagger category is dagger complete (admits all dagger limits of a given shape) if and only if it has dagger equalizers, dagger products, and dagger intersections.
• Dagger limits can be characterized as right adjoints to the diagonal functor in the dagger 2-category of dagger categories, extending the classical adjoint functor theory.
• In categories with polar decomposition, ordinary limits can be upgraded to dagger limits via the dagger structure, providing a constructive method.
• Dagger limits commute with dagger colimits if and only if the bifunctor $ D: f{J} imes f{K} o f{C} $ is adjointable, and in this case the canonical comparison morphism is unitary.
• Counterexamples show that without adjointability, the order of taking limits and colimits may not commute, even non-unitarily, as demonstrated in the category of finite-dimensional Hilbert spaces.