Unitary, Inner product, and Dagger categories

Abstract: This article provides an alternate characterization of dagger categories, which are central to the study of categorical quantum mechanics, in terms of inner product categories. An inner product category is an "achiral involutive" category with an inner product combinator. Inner product categories are, in turn, precisely the same as unitary categories, which are a weaker form of dagger categories. In unitary categories, there is an isomorphism between an object and its dagger, instead of the identity function as in the case of dagger categories. Every unitary category is equipped with a global inner product structure, which allows one to strictify the involutive structure on the unitary category to obtain a dagger category, making unitary categories 2-categorically equivalent to dagger categories. By regarding the inner product as an abstract metric on an (achiral) involutive category, one can define metric-preserving maps (isometries) in inner product categories, and also develop other notions of special maps -- unitary, Hermitian, positive, and normal maps -- in this setting.

• The paper proves a three-way 2-categorical equivalence between unitary categories, inner product categories, and dagger categories, showing that dagger strictness can be recovered from weaker object-wise duality data.
• The authors construct inner products from unitary structure maps and recover those maps from inner products, while deriving inner adjoints that define a genuine strict dagger structure on the same category.
• The framework characterizes isometries, unitary, Hermitian, positive, and normal maps, but remains limited to achiral involutions and leaves chiral extensions and dagger-linearly distributive applications for future work.

This paper develops a categorical framework in which dagger categories are characterized, up to 2-categorical equivalence, by two weaker structures: unitary categories and inner product categories. The authors — Cockett, Kumar, and Srinivasan (2603.27614) — build on Mellies' notion of chirality for involutions and on prior work of Srinivasan on dagger isomix categories, replacing the strict identity-on-objects involution of Selinger's dagger categories with an involutor natural isomorphism $\iota_X : X \to X^{\dagger\dagger}$.

Achiral involutive categories

The base structure is an achiral involutive category: a category $\mathbb{X}$ with a contravariant endofunctor $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$ and a natural isomorphism $\iota$ (the involutor) satisfying the coherence condition $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$. This coherence makes $(\_)^\dagger$ part of an adjoint equivalence between $\mathbb{X}$ and $\mathbb{X}^{\mathrm{op}}$, though the authors note that most of their development does not require this identity. Strict involutive categories — where $X^\dagger = X$ and $\iota = \mathrm{id}$ — coincide exactly with dagger categories.

Examples include all dagger categories (groupoids, inverse categories, relations, matrices over rigs with conjugation), $\mathbb{X}$0-autonomous and compact closed categories with their dual functors, symmetric reflexive graphs, and posets such as the integers under negation. The last two examples are emphatically not dagger categories, illustrating why the relaxed notion is needed: as the authors observe, $\mathbb{X}$1-autonomous and compact closed categories can have necessarily non-strict involutions.

The paper then equips these categories with a 2-categorical structure. An involutive functor $\mathbb{X}$2 carries a preservator $\mathbb{X}$3 subject to a coherence square interacting with the involutors; notably, unlike the covariant case where coherence forces the preservator to be invertible automatically, here invertibility must be assumed. Achiral transformations are natural transformations satisfying a "tethering" condition involving a map running in the reverse direction. A key structural result is that tethering forces every achiral transformation to be a natural isomorphism, proved by showing $\mathbb{X}$4 is simultaneously a section and a retraction via conjugation arguments. In the strict case this recovers Selinger's unitary natural transformations ($\mathbb{X}$5). These data organize into a 2-category aChiral in which all 2-cells are isomorphisms.

Unitary categories

A unitary category weakens the dagger condition by requiring only that each object admit an isomorphism $\mathbb{X}$6 (the unitary structure map, not required to be natural) satisfying:

• [U.1] $\mathbb{X}$7
• [U.2] $\mathbb{X}$8

Every dagger category is unitary with $\mathbb{X}$9, and finite-dimensional vector spaces form the standard example: each space is isomorphic to its dual, but unnaturally (via a choice of basis). The integers under negation give a simple non-unitary example.

Unitary functors satisfy $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$0; two consequences follow immediately. First, the preservator is determined by the underlying functor, $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$1, so the earlier assumption of invertibility becomes automatic. Second, a unitary functor between dagger categories preserves the dagger strictly on the nose. Unitary categories, unitary functors, and transformations form a 2-category UNITARY.

The central construction extracts from any involutive category $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$2 its full subcategory $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$3 of pre-unitary objects $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$4 with $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$5. The computation is instructive: in $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$6 the pre-unitary objects reduce to the empty relation on the null set and the unique reflexive relation on a singleton; in the integer poset one obtains the one-object discrete category, and the nonzero integers have no pre-unitary objects at all. For a dagger category, pre-unitary objects are precisely Hermitian isomorphisms. The assignment extends to a 2-functor, and the authors establish a 2-adjunction

$$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$7

with $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$8, making unitary categories a 2-coreflective subcategory of achiral involutive categories.

Inner product categories and the three-way equivalence

An inner product category is an involutive category equipped with an inner product combinator assigning to $$(\_)^\dagger : \mathbb{X}^{\mathrm{op}} \to \mathbb{X}$$9 and $\iota$0 a map $\iota$1, subject to four axioms governing pre-composition, interaction with dagger and duals, and cancellation against split epis/monos. Unlike Vicary's inner product for parallel maps in a dagger category, this combinator applies to maps with arbitrary common codomain; it reduces to Vicary's definition when the maps are parallel and the involution strict.

The main theorem is a three-way equivalence of structures:

1. Every unitary category is an inner product category via $\iota$2.
2. Conversely, every inner product category is unitary with unitary structure map $\iota$3, whose inverse is the dual inner product $\iota$4.
3. These correspondences are bijective, and extend to functors: an involutive functor preserves the inner product if and only if it is a unitary functor.

A notable technical observation is that axiom IP.2 is redundant given IP.2 together with the double-dagger axiom IP.2 — Proposition establishing the unitary structure from the inner product never uses (b), and the redundancy is proved explicitly in the appendix.

Inner adjoints and strictification

Given the inner product, each map $\iota$5 admits a unique inner adjoint $\iota$6 characterized by $\iota$7. The adjoint operation is contravariant, stationary on objects, involutive ($\iota$8), and reverses composition. Consequently, every inner product category induces a genuine dagger category $\iota$9 — the adjoint involution — and the identity functor with preservator $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$0 is a unitary equivalence between $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$1 and $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$2. This yields the paper's headline result: the 2-category of unitary (equivalently, inner product) categories is equivalent to the 2-category of dagger categories. In other words, the strictness built into Selinger's definition is not a loss of generality at the level of 2-categories: any unitary category can be strictified to a dagger category using its global inner product.

Isometries and special maps

Treating the inner product as an abstract metric, the paper defines and characterizes several classes of maps. An isometry $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$3 satisfies $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$4 for all $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$5, equivalently $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$6, equivalently $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$7. Isomorphisms that are isometries are unitary maps, characterized by $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$8 among other conditions. Both the unitary structure map $\iota_{X^\dagger}(\iota_X)^\dagger = 1_{X^\dagger}$9 and the involutor $(\_)^\dagger$0 are shown to be isometries (hence unitary). Hermitian maps satisfy $(\_)^\dagger$1, equivalently $(\_)^\dagger$2; positive maps are those of the form $(\_)^\dagger$3 (reducing to the standard dagger-category notion $(\_)^\dagger$4); normal maps satisfy $(\_)^\dagger$5. Containment results show every Hermitian map and every unitary endomorphism is normal.

Limitations and open questions

Several caveats bear directly on the results. First, the tethering requirement makes all achiral transformations isomorphisms, which is restrictive; the authors state that a less drastic chiral formulation exists and "has interesting structural ramifications," but defer it to a sequel. Second, the preservator of an involutive functor must be assumed invertible — unlike the covariant setting, where coherence suffices — and the authors do not offer a proof that this assumption can be dropped. Third, the framework covers only the achiral case: chiral involutive categories are not unitary and carry no inner product, so the entire equivalence machinery does not apply there. Finally, the connection to $(\_)^\dagger$6-linearly distributive categories, whose models study infinite-dimensional quantum processes, is announced but not developed in this paper.

Conclusion

The paper establishes that dagger categories, unitary categories, and inner product categories are 2-categorically equivalent, with unitary categories forming a 2-coreflection inside achiral involutive categories. The practical consequence is that the strictness demanded by the classical definition of a dagger category can always be recovered from weaker, more flexible data — an object-wise isomorphism to the dagger dual plus a global inner product — while the inner product simultaneously supports Hilbert-space-style notions of adjoint, isometry, unitary, Hermitian, positive, and normal maps. The chiral generalization and its link to $(\_)^\dagger$7-linearly distributive categories remain open in this work.