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Copy file name to clipboardExpand all lines: docs/src/appendix/categories.md
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@@ -27,9 +27,7 @@ To morphisms, we associate boxes with an incoming and outgoing line denoting the
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The flow from source to target, and thus the direction of morphism composition ``f ∘ g`` (sometimes known as the flow of time) can be chosen left to right (like the arrow in ``f:W→V``), right to left (like the composition order ``f ∘ g``, or the matrix product), bottom to top (quantum field theory convention) or top to bottom (quantum circuit convention).
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Throughout this manual, we stick to this latter convention (which is not very common in manuscripts on category theory):
The direction of the arrows, which become important once we introduce duals, are also subject to convention, and are here chosen to follow the arrow in ``f:W→V``, i.e. the source comes in and the target goes out.
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Strangely enough, this is opposite to the most common convention.
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Note that any consistent way of inserting the associator or left or right unitor to convert a graphical representation to a diagram of compositions and tensor products of morphisms gives rise to the same result, by virtue of Mac Lane's coherence theorem.
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Using the horizontal direction (left to right) to stack tensor products, this gives rise to the following graphical notation for the tensor product of two morphisms, and for a general morphism ``t`` between a tensor product of objects in source and target:
Another relevant example is the category ``\mathbf{SVect}_𝕜``, which has as objects *super vector spaces* over ``𝕜``, which are vector spaces with a ``ℤ₂`` grading, i.e. they are decomposed as a direct sum ``V = V_0 ⊕ V_1``.
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Furthermore, the morphisms between two super vector spaces are restricted to be grading preserving, i.e. ``f∈ \mathrm{Hom}_{\mathbf{SVect}}(W,V)`` has ``f(W_0) ⊂ V_0`` and ``f(W_1) ⊂ V_1``.
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Graphically, we represent the exact pairing and snake rules as
Note that we denote the dual objects ``{}^{∨}V`` as a line ``V`` with arrows pointing in the opposite (i.e. upward) direction.
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This notation is related to quantum field theory, where anti-particles are (to some extent) interpreted as particles running backwards in time.
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These exact pairings are known as the left evaluation and coevaluation, and ``{}^{∨}V`` is the left dual of ``V``.
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Likewise, we can also define a right dual ``V^{∨}`` of ``V`` and associated pairings, the right evaluation ``\tilde{ϵ}_V: V ⊗ V^{∨} → I`` and coevaluation ``\tilde{η}_V: I → V^{∨} ⊗ V``, satisfying
In particular, one could choose ``\tilde{ϵ}_{{}^{∨}V} = ϵ_V`` and thus define ``V`` as the right dual of ``{}^{∨}V``.
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While there might be other choices, this choice must at least be isomorphic, such that ``({}^{∨}V)^{∨} ≂ V``.
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If objects ``V`` and ``W`` have left (respectively right) duals, then for a morphism ``f ∈ \mathrm{Hom}(W,V)``, we furthermore define the left (respectively right) *transpose*``{}^{∨}f ∈ \mathrm{Hom}({}^{∨}V, {}^{∨}W)`` (respectively ``f^{∨} ∈ \mathrm{Hom}(V^{∨}, W^{∨})``) as
where on the right we also illustrate the mapping from ``t ∈ \mathrm{Hom}(W_1 ⊗ W_2 ⊗ W_3, V_1 ⊗ V_2)`` to a morphism in ``\mathrm{Hom}(I, V_1 ⊗ V_2 ⊗ {}^{∨} W_3 ⊗ {}^{∨} W_2 ⊗ {}^{∨} W_1)``.
Note that ``\mathrm{tr}_{\mathrm{l}}(f) = \mathrm{tr}_{\mathrm{r}}(f*)`` and that ``\mathrm{tr}_{\mathrm{l}/\mathrm{r}}(f∘g) = \mathrm{tr}_{\mathrm{l}/\mathrm{r}}(g∘f)``.
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The braiding isomorphism ``τ_{V,W}`` and its inverse are graphically represented as the lines ``V`` and ``W`` crossing over and under each other:
where the expression on the right hand side, ``τ_{W,V}∘τ_{V,W}`` can generically not be simplified.
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Hence, for general braidings, there is no unique choice to identify a tensor in ``V⊗W`` and ``W⊗V``, as the isomorphisms ``τ_{V,W}``, ``τ_{W,V}^{-1}``, ``τ_{V,W} ∘ τ_{W,V} ∘ τ_{V,W}``, … mapping from ``V⊗W`` to ``W⊗V`` can all be different.
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The braiding of a space and a dual space also follows naturally, it is given by ``τ_{V^*,W} = λ_{W ⊗ V^*} ∘ (ϵ_V ⊗ \mathrm{id}_{W ⊗ V^*}) ∘ (\mathrm{id}_{V^*} ⊗ τ_{V,W}^{-1} ⊗ \mathrm{id}_{V^*}) ∘ (\mathrm{id}_{V^*⊗ W} ⊗ η_V) ∘ ρ_{V^* ⊗ W}^{-1}``, i.e.
**Balanced categories**``C`` are braided categories that come with a **twist**``θ``, a natural transformation from the identity functor ``1_C`` to itself, such that ``θ_V ∘ f = f ∘ θ_W`` for all morphisms ``f ∈ \mathrm{Hom}(W,V)``, and for which the main requirement is that
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where we omitted the necessary left and right unitors and associators.
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Graphically, the twists and their inverse (for which we refer to [^Turaev]) are then represented as
The graphical representation also makes it straightforward to verify that ``(θ^{\mathrm{l}}_V)^* = θ^{\mathrm{r}}_{V^*}``, ``(θ^{\mathrm{r}}_V)^* = θ^{\mathrm{l}}_{V^*}`` and ``\mathrm{tr}_{\mathrm{l}}( θ^{\mathrm{r}}_V ) = \mathrm{tr}_{\mathrm{r}}( θ^{\mathrm{l}}_V )``.
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or graphically
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```@raw html
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<img src="../img/diagram-pivotalfromtwist.svg" alt="pivotal from twist" class="color-invertible"/>
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```
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where we have drawn ``θ`` as ``θ^{\mathrm{l}}`` on the left and as ``θ^{\mathrm{r}}`` on the right, but in this case the starting assumption was that they are one and the same, and we defined the pivotal structure so as to make it compatible with the graphical representation.
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This construction of the pivotal structure can than be used to define the trace, which is spherical, i.e.
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In the graphical representation, the dagger of a morphism can be represented by mirroring the morphism around a horizontal axis, and then reversing all arrows (bringing them back to their original orientation before the mirror operation):
and also refer to the inclusion and projection maps as splitting and fusion tensor, respectively.
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@@ -524,18 +498,14 @@ which defines the *F-symbol*, i.e. the matrix elements of the associator
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Note that the left hand side represents a map in ``\mathrm{Hom}(d′,d)``, which must be zero if ``d′`` is different from ``d``, hence the ``δ_{d,d′}`` on the right hand side.
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In a strict category, or in the graphical notation, the associator ``α`` is omitted and these relations thus represent a unitary basis transform between the basis of inclusion maps ``X_{d,(eμν)}^{abc}`` and ``\tilde{X}_{d,(fκλ)}^{abc}``, which is also called an F-move, i.e. graphically:
The matrix ``F^{abc}_d`` is thus a unitary matrix.
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The pentagon coherence equation can also be rewritten in terms of these matrix elements, and as such yields the celebrated pentagon equation for the F-symbols.
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In a similar fashion, the unitors result in ``N^{a1}_{b} = N^{1a}_b = δ^{a}_b`` (where we have now written ``1`` instead of ``I`` for the unit object) and the triangle equation leads to additional relations between the F- symbols involving the unit object.
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In particular, if we identify ``X^{1a}_{a,1}:a→(1⊗a)`` with ``λ_a^†`` and ``X^{a1}_{a,1}:a→(a⊗1)`` with ``ρ_a^†``, the triangle equation and its collaries imply that ``[F^{1ab}_{c}]_{(11μ)}^{(cν1)} = δ^{ν}_{μ}``, and similar relations for ``F^{a1b}_c`` and ``F^{ab1}_c``, which are graphically represented as
In the case of group representations, i.e. the category ``\mathbf{Rep}_{\mathsf{G}}``, the splitting and fusion tensors are known as the Clebsch-Gordan coefficients, especially in the case of ``\mathsf{SU}_2``.
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An F-move amounts to a recoupling and the F-symbols can thus be identified with the *6j-symbols* (strictly speaking, Racah's W-symbol for ``\mathsf{SU}_2``).
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However, for ``a=\bar{a}``, the value of ``χ_a`` cannot be changed, but must satisfy ``χ_a^2 = 1``, or thus ``χ_a = ±1``.
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This value is a topological invariant known as the *Frobenius-Schur indicator*. Graphically, we represent this isomorphism and its relations as
where again bar denotes complex conjugation in the second line, and we introduced two new families of matrices ``A^{ab}_c`` and ``B^{ab}_c``, whose entries are composed out of entries of the F-symbol, namely
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Composing the left hand side of first graphical equation with its dagger, and noting that the resulting element ``f ∈ \mathrm{End}(a)`` must satisfy ``f = d_a^{-1} \mathrm{tr}(f) \mathrm{id}_a``, i.e.
Henceforth, we reserve ``θ_a`` for the scalar value itself. Note that ``θ_a = θ_{\bar{a}}`` as our category is spherical and thus a ribbon category, and that the defining relation of a twist implies
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If ``a = \bar{a}``, we can furthermore relate the twist, the braiding and the Frobenius- Schur indicator via ``θ_a χ_a R^{aa}_1 =1``, because of
For the recurring example of ``\mathbf{Rep}_{\mathsf{G}}``, the braiding acts simply as the swap of the two vector spaces on which the representations are acting and is thus symmetric, i.e. ``τ_{b,a} ∘ τ_{a,b} = \mathrm{id}_{a⊗b}``.
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