In the dagger category FHilb, suppose f: ℂ² → ℂ³ is a linear map. What is the type of its dagger f†?
Af†: ℂ² → ℂ³ — the dagger preserves morphism direction and fixes both objects
Bf†: ℂ³ → ℂ² — the dagger reverses morphism direction while fixing objects
Cf† does not exist because f is not invertible (ℂ² and ℂ³ have different dimensions)
Df†: ℂ³ → ℂ³ — the dagger produces a unitary map on the codomain
The dagger reverses morphism direction while fixing objects. If f: A → B, then f†: B → A. In FHilb, this is the adjoint (conjugate transpose): a matrix from ℂ² to ℂ³ is a 3×2 matrix, and its conjugate transpose is a 2×3 matrix, i.e., a map from ℂ³ to ℂ². The dagger does NOT require f to be invertible — it exists for all morphisms. Option A confuses the dagger with the identity; option C confuses the dagger with the categorical inverse.
Question 2 Multiple Choice
Which of the following correctly describes 'unitary' morphisms in a dagger category?
AA morphism f: A → B is unitary if f† = f (the morphism equals its own dagger)
BA morphism f: A → B is unitary if f† ∘ f = id_A and f ∘ f† = id_B — the dagger is a two-sided inverse
CA morphism is unitary if and only if it is an isomorphism (has an ordinary inverse)
DA unitary morphism is one where f† is defined; all morphisms in a dagger category are automatically unitary
A unitary morphism satisfies f† ∘ f = id_A and f ∘ f† = id_B — the dagger serves as a two-sided inverse. This is the categorical generalization of unitary matrices (length-preserving isometries). Note that option A describes self-adjoint (Hermitian) morphisms: f = f†. Unitary and self-adjoint are distinct classes. Not all isomorphisms are unitary — an isomorphism has a categorical inverse, but the dagger provides a geometrically meaningful inverse that respects inner product structure, not just set-theoretic structure.
Question 3 True / False
In a dagger category, nearly every morphism f: A → B has a two-sided categorical inverse given by f†.
TTrue
FFalse
Answer: False
The dagger f†: B → A is always defined (for every morphism), but it is not always a categorical inverse. For f† to be a two-sided inverse, you would need f† ∘ f = id_A and f ∘ f† = id_B — this is the definition of a unitary morphism, and only unitaries have this property. A non-unitary morphism like a projection or an embedding has a well-defined dagger, but f† ∘ f ≠ id in general. For example, in FHilb the orthogonal projection onto a subspace has a dagger (itself), but is not invertible.
Question 4 True / False
The dagger structure of a dagger category is additional data — it cannot be recovered from the category's objects and morphisms alone.
TTrue
FFalse
Answer: True
This is stated explicitly in the definition: a dagger category is a category *equipped with* a choice of dagger functor †. The same underlying category can in principle carry different dagger structures, or none at all. In FHilb, the dagger is the adjoint operator, but this requires knowing which maps are 'adjoint-compatible' — information that lives outside the purely set-theoretic composition structure. This is analogous to how a group can carry different group structures; the algebraic data is additional, not inherent.
Question 5 Short Answer
What makes the dagger operation different from taking the ordinary categorical inverse of a morphism, and why does this distinction matter?
Think about your answer, then reveal below.
Model answer: The ordinary categorical inverse f⁻¹ of a morphism f: A → B (when it exists) satisfies f⁻¹ ∘ f = id_A and f ∘ f⁻¹ = id_B, and it exists only for isomorphisms. The dagger f†: B → A always exists (for every morphism in a dagger category) but need not be an inverse — f† ∘ f is a morphism A → A that may be the identity (for unitaries) or may not (for non-isometric maps). The distinction matters because f† carries geometric meaning (transposing/adjoint) even when f is not invertible, enabling the definition of self-adjoint observables and unitary quantum gates in categorical quantum mechanics.
In linear algebra terms: a non-square matrix has no inverse, but always has a conjugate transpose (adjoint). The adjoint captures inner-product-respecting structure even when full invertibility fails. Categorically, the dagger generalizes this — it makes sense in any category where morphisms have a natural 'reversal' that respects some additional structure (like inner products), without requiring those morphisms to be isomorphisms. This is why dagger categories are the right framework for quantum mechanics, where projections and partial isometries are central objects.