@@ -46,7 +46,9 @@ matrix product), bottom to top (quantum field theory convention) or top to botto
4646circuit convention). Throughout this manual, we stick to this latter convention (which is
4747not very common in manuscripts on category theory):
4848
49- ![ composition] ( img/diagram_morphism.svg )
49+ ``` @raw html
50+ <img src="../img/diagram_morphism.svg" alt="composition" class="color-invertible"/>
51+ ```
5052
5153The direction of the arrows, which become important once we introduce duals, are also
5254subject to convention, and are here chosen to follow the arrow in `` f:W→V `` , i.e. the
@@ -168,7 +170,9 @@ this gives rise to the following graphical notation for the tensor product of tw
168170morphisms, and for a general morphism `` t `` between a tensor product of objects in source
169171and target:
170172
171- ![ tensorproduct] ( img/diagram-tensorproduct.svg )
173+ ``` @raw html
174+ <img src="../img/diagram-tensorproduct.svg" alt="tensorproduct" class="color-invertible"/>
175+ ```
172176
173177Another relevant example is the category `` \mathbf{SVect}_𝕜 `` , which has as objects * super
174178vector spaces* over `` 𝕜 `` , which are vector spaces with a `` ℤ₂ `` grading, i.e.
@@ -258,7 +262,9 @@ coevaluation of ``V`` and ``W``, such that ``{}^{∨}W ⊗ {}^{∨}V`` is at lea
258262
259263Graphically, we represent the exact pairing and snake rules as
260264
261- ![ left dual] ( img/diagram-leftdual.svg )
265+ ``` @raw html
266+ <img src="../img/diagram-leftdual.svg" alt="left dual" class="color-invertible"/>
267+ ```
262268
263269Note that we denote the dual objects `` {}^{∨}V `` as a line `` V `` with arrows pointing in the
264270opposite (i.e. upward) direction. This notation is related to quantum field theory, where
@@ -269,7 +275,9 @@ the left dual of ``V``. Likewise, we can also define a right dual ``V^{∨}`` of
269275associated pairings, the right evaluation `` \tilde{ϵ}_V: V ⊗ V^{∨} → I `` and coevaluation
270276`` \tilde{η}_V: I → V^{∨} ⊗ V `` , satisfying
271277
272- ![ right dual] ( img/diagram-rightdual.svg )
278+ ``` @raw html
279+ <img src="../img/diagram-rightdual.svg" alt="right dual" class="color-invertible"/>
280+ ```
273281
274282In particular, one could choose `` \tilde{ϵ}_{{}^{∨}V} = ϵ_V `` and thus define `` V `` as the
275283right dual of `` {}^{∨}V `` . While there might be other choices, this choice must at least be
@@ -278,7 +286,9 @@ isomorphic, such that ``({}^{∨}V)^{∨} ≂ V``.
278286If objects `` V `` and `` W `` have left (respectively right) duals, than for a morphism `` f ∈ \mathrm{Hom}(W,V) `` , we furthermore define the left (respectively right)
279287* transpose* `` {}^{∨}f ∈ \mathrm{Hom}({}^{∨}V, {}^{∨}W) `` (respectively `` f^{∨} ∈ \mathrm{Hom}(V^{∨}, W^{∨}) `` ) as
280288
281- ![ transpose] ( img/diagram-transpose.svg )
289+ ``` @raw html
290+ <img src="../img/diagram-transpose.svg" alt="transpose" class="color-invertible"/>
291+ ```
282292
283293where on the right we also illustrate the mapping from
284294`` t ∈ \mathrm{Hom}(W_1 ⊗ W_2 ⊗ W_3, V_1 ⊗ V_2) `` to a morphism in
@@ -339,7 +349,9 @@ and a right trace as
339349
340350They are graphically represented as
341351
342- ![ trace] ( img/diagram-trace.svg )
352+ ``` @raw html
353+ <img src="../img/diagram-trace.svg" alt="trace" class="color-invertible"/>
354+ ```
343355
344356and they do not need to coincide. Note that
345357`` \mathrm{tr}_{\mathrm{l}}(f) = \mathrm{tr}_{\mathrm{r}}(f*) `` and that
@@ -391,11 +403,15 @@ has been omitted). We also have ``λ_V ∘ τ_{V,I} = ρ_{V,I}``, ``ρ_V ∘ τ_
391403The braiding isomorphism `` τ_{V,W} `` and its inverse are graphically represented as the
392404lines `` V `` and `` W `` crossing over and under each other:
393405
394- ![ braiding] ( img/diagram-braiding.svg )
406+ ``` @raw html
407+ <img src="../img/diagram-braiding.svg" alt="braiding" class="color-invertible"/>
408+ ```
395409
396410such that we have
397411
398- ![ braiding relations] ( img/diagram-braiding2.svg )
412+ ``` @raw html
413+ <img src="../img/diagram-braiding2.svg" alt="braiding relations" class="color-invertible"/>
414+ ```
399415
400416where the expression on the right hand side, `` τ_{W,V}∘τ_{V,W} `` can generically not be
401417simplified. Hence, for general braidings, there is no unique choice to identify a tensor in
@@ -420,7 +436,9 @@ of ``\mathbf{SVect}``, which will again be studied in detail at the end of this
420436The braiding of a space and a dual space also follows naturally, it is given by
421437`` 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.
422438
423- ![ braiding dual] ( img/diagram-braidingdual.svg )
439+ ``` @raw html
440+ <img src="../img/diagram-braidingdual.svg" alt="braiding dual" class="color-invertible"/>
441+ ```
424442
425443** Balanced categories** `` C `` are braided categories that come with a ** twist** `` θ `` , a
426444natural transformation from the identity functor `` 1_C `` to itself, such that
441459where we omitted the necessary left and right unitors and associators. Graphically, the
442460twists and their inverse (for which we refer to [ ^ turaev ] ) are then represented as
443461
444- ![ twists] ( img/diagram-twists.svg )
462+ ``` @raw html
463+ <img src="../img/diagram-twists.svg" alt="twists" class="color-invertible"/>
464+ ```
445465
446466The graphical representation also makes it straightforward to verify that
447467`` (θ^{\mathrm{l}}_V)^* = θ^{\mathrm{r}}_{V^*} `` ,
@@ -465,7 +485,9 @@ structure, or, to define the exact pairing for the right dual functor as
465485
466486or graphically
467487
468- ![ pivotal from twist] ( img/diagram-pivotalfromtwist.svg )
488+ ``` @raw html
489+ <img src="../img/diagram-pivotalfromtwist.svg" alt="pivotal from twist" class="color-invertible"/>
490+ ```
469491
470492where we have drawn `` θ `` as `` θ^{\mathrm{l}} `` on the left and as `` θ^{\mathrm{r}} `` on
471493the right, but in this case the starting assumption was that they are one and the same, and
@@ -520,7 +542,9 @@ In the graphical representation, the dagger of a morphism can be represented by
520542the morphism around a horizontal axis, and then reversing all arrows (bringing them back to
521543their original orientation before the mirror operation):
522544
523- ![ dagger] ( img/diagram-dagger.svg )
545+ ``` @raw html
546+ <img src="../img/diagram-dagger.svg" alt="dagger" class="color-invertible"/>
547+ ```
524548
525549where for completeness we have also depicted the graphical representation of the transpose,
526550which is a very different operation. In particular, the dagger does not reverse the order
@@ -704,7 +728,9 @@ fusion category, on which we now focus, the corresponding projection maps are
704728
705729Graphically, we represent these relations as
706730
707- ![ fusion] ( img/diagram-fusion.svg )
731+ ``` @raw html
732+ <img src="../img/diagram-fusion.svg" alt="fusion" class="color-invertible"/>
733+ ```
708734
709735and also refer to the inclusion and projection maps as splitting and fusion tensor,
710736respectively.
@@ -744,7 +770,9 @@ thus represent a unitary basis transform between the basis of inclusion maps
744770`` X_{d,(eμν)}^{abc} `` and `` \tilde{X}_{d,(fκλ)}^{abc} `` , which is also called an F-move,
745771i.e. graphically:
746772
747- ![ Fmove] ( img/diagram-Fmove.svg )
773+ ``` @raw html
774+ <img src="../img/diagram-Fmove.svg" alt="Fmove" class="color-invertible"/>
775+ ```
748776
749777The matrix `` F^{abc}_d `` is thus a unitary matrix. The pentagon coherence equation can also
750778be rewritten in terms of these matrix elements, and as such yields the celebrated pentagon
@@ -757,7 +785,9 @@ triangle equation and its collaries imply that
757785`` [F^{1ab}_{c}]_{(11μ)}^{(cν1)} = δ^{ν}_{μ} `` , and similar relations for `` F^{a1b}_c `` and
758786`` F^{ab1}_c `` , which are graphically represented as
759787
760- ![ Fmove1] ( img/diagram-Fmove1.svg )
788+ ``` @raw html
789+ <img src="../img/diagram-Fmove1.svg" alt="Fmove1" class="color-invertible"/>
790+ ```
761791
762792In the case of group representations, i.e. the category `` \mathbf{Rep}_{\mathsf{G}} `` , the
763793splitting and fusion tensors are known as the Clebsch-Gordan coefficients, especially in
@@ -803,7 +833,9 @@ for ``a=\bar{a}``, the value of ``χ_a`` cannot be changed, but must satisfy ``
803833or thus `` χ_a = ±1 `` . This value is a topological invariant known as the
804834* Frobenius-Schur indicator* . Graphically, we represent this isomorphism and its relations as
805835
806- ![ Zisomorphism] ( img/diagram-Zisomorphism.svg )
836+ ``` @raw html
837+ <img src="../img/diagram-Zisomorphism.svg" alt="Zisomorphism" class="color-invertible"/>
838+ ```
807839
808840We can now discuss the relation between the exact pairing and the fusion and splitting
809841tensors. Given that the (left) coevaluation `` η_a ∈ \mathrm{Hom}(1, a⊗a^*) `` , we can define the
@@ -822,12 +854,16 @@ snake rules. Hence, both the quantum dimensions and the Frobenius-Schur indicato
822854encoded in the F-symbol. Hence, they do not represent new independent data. Again, the
823855graphical representation is more enlightning:
824856
825- ![ ZtoF] ( img/diagram-ZtoF.svg )
857+ ``` @raw html
858+ <img src="../img/diagram-ZtoF.svg" alt="ZtoF" class="color-invertible"/>
859+ ```
826860
827861With these definitions, we can now also evaluate the action of the evaluation map on the
828862splitting tensors, namely
829863
830- ![ splittingfusionrelation] ( img/diagram-splittingfusionrelation.svg )
864+ ``` @raw html
865+ <img src="../img/diagram-splittingfusionrelation.svg" alt="splittingfusionrelation" class="color-invertible"/>
866+ ```
831867
832868where again bar denotes complex conjugation in the second line, and we introduced two new
833869families of matrices `` A^{ab}_c `` and `` B^{ab}_c `` , whose entries are composed out of
@@ -843,7 +879,9 @@ Composing the left hand side of first graphical equation with its dagger, and no
843879the resulting element `` f ∈ \mathrm{End}(a) `` must satisfy
844880`` f = d_a^{-1} \mathrm{tr}(f) \mathrm{id}_a `` , i.e.
845881
846- ![ Brelation] ( img/diagram-Brelation.svg )
882+ ``` @raw html
883+ <img src="../img/diagram-Brelation.svg" alt="Brelation" class="color-invertible"/>
884+ ```
847885
848886allows to conclude that
849887`` ∑_ν [B^{ab}_c]^{ν}_{μ} \overline{[B^{ab}_c]^{ν}_{μ′}} = \delta_{μ,μ′} `` , i.e. `` B^{ab}_c ``
@@ -893,7 +931,9 @@ the simple objects. We can then express ``τ_{a,b}`` in terms of its matrix elem
893931
894932or graphically
895933
896- ![ braidingR] ( img/diagram-braidingR.svg )
934+ ``` @raw html
935+ <img src="../img/diagram-braidingR.svg" alt="braidingR" class="color-invertible"/>
936+ ```
897937
898938The hexagon coherence axiom for the braiding and the associator can then be reexpressed in
899939terms of the F-symbols and R-symbols.
@@ -905,7 +945,9 @@ complex phases because of unitarity) multiplying the identity morphism, i.e.
905945
906946or graphically
907947
908- ![ simpletwist] ( img/diagram-simpletwist.svg )
948+ ``` @raw html
949+ <img src="../img/diagram-simpletwist.svg" alt="simpletwist" class="color-invertible"/>
950+ ```
909951
910952Henceforth, we reserve `` θ_a `` for the scalar value itself. Note that `` θ_a = θ_{\bar{a}} ``
911953as our category is spherical and thus a ribbon category, and that the defining relation of
@@ -916,7 +958,9 @@ a twist implies
916958If `` a = \bar{a} `` , we can furthermore relate the twist, the braiding and the Frobenius-
917959Schur indicator via `` θ_a χ_a R^{aa}_1 =1 `` , because of
918960
919- ![ twistfrobeniusschur] ( img/diagram-twistfrobeniusschur.svg )
961+ ``` @raw html
962+ <img src="../img/diagram-twistfrobeniusschur.svg" alt="twistfrobeniusschur" class="color-invertible"/>
963+ ```
920964
921965For the recurring example of `` \mathbf{Rep}_{\mathsf{G}} `` , the braiding acts simply as the
922966swap of the two vector spaces on which the representations are acting and is thus symmetric,
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