@@ -47,7 +47,7 @@ circuit convention). Throughout this manual, we stick to this latter convention
4747not very common in manuscripts on category theory):
4848
4949``` @raw html
50- <img src="img/diagram_morphism.svg" alt="composition" class="color-invertible"/>
50+ <img src="../ img/diagram_morphism.svg" alt="composition" class="color-invertible"/>
5151```
5252
5353The direction of the arrows, which become important once we introduce duals, are also
@@ -171,7 +171,7 @@ morphisms, and for a general morphism ``t`` between a tensor product of objects
171171and target:
172172
173173``` @raw html
174- <img src="img/diagram-tensorproduct.svg" alt="tensorproduct" class="color-invertible"/>
174+ <img src="../ img/diagram-tensorproduct.svg" alt="tensorproduct" class="color-invertible"/>
175175```
176176
177177Another relevant example is the category `` \mathbf{SVect}_𝕜 `` , which has as objects * super
@@ -263,7 +263,7 @@ coevaluation of ``V`` and ``W``, such that ``{}^{∨}W ⊗ {}^{∨}V`` is at lea
263263Graphically, we represent the exact pairing and snake rules as
264264
265265``` @raw html
266- <img src="img/diagram-leftdual.svg" alt="left dual" class="color-invertible"/>
266+ <img src="../ img/diagram-leftdual.svg" alt="left dual" class="color-invertible"/>
267267```
268268
269269Note that we denote the dual objects `` {}^{∨}V `` as a line `` V `` with arrows pointing in the
@@ -276,7 +276,7 @@ associated pairings, the right evaluation ``\tilde{ϵ}_V: V ⊗ V^{∨} → I``
276276`` \tilde{η}_V: I → V^{∨} ⊗ V `` , satisfying
277277
278278``` @raw html
279- <img src="img/diagram-rightdual.svg" alt="right dual" class="color-invertible"/>
279+ <img src="../ img/diagram-rightdual.svg" alt="right dual" class="color-invertible"/>
280280```
281281
282282In particular, one could choose `` \tilde{ϵ}_{{}^{∨}V} = ϵ_V `` and thus define `` V `` as the
@@ -287,7 +287,7 @@ If objects ``V`` and ``W`` have left (respectively right) duals, than for a morp
287287* transpose* `` {}^{∨}f ∈ \mathrm{Hom}({}^{∨}V, {}^{∨}W) `` (respectively `` f^{∨} ∈ \mathrm{Hom}(V^{∨}, W^{∨}) `` ) as
288288
289289``` @raw html
290- <img src="img/diagram-transpose.svg" alt="transpose" class="color-invertible"/>
290+ <img src="../ img/diagram-transpose.svg" alt="transpose" class="color-invertible"/>
291291```
292292
293293where on the right we also illustrate the mapping from
@@ -350,7 +350,7 @@ and a right trace as
350350They are graphically represented as
351351
352352``` @raw html
353- <img src="img/diagram-trace.svg" alt="trace" class="color-invertible"/>
353+ <img src="../ img/diagram-trace.svg" alt="trace" class="color-invertible"/>
354354```
355355
356356and they do not need to coincide. Note that
@@ -404,13 +404,13 @@ The braiding isomorphism ``τ_{V,W}`` and its inverse are graphically represente
404404lines `` V `` and `` W `` crossing over and under each other:
405405
406406``` @raw html
407- <img src="img/diagram-braiding.svg" alt="braiding" class="color-invertible"/>
407+ <img src="../ img/diagram-braiding.svg" alt="braiding" class="color-invertible"/>
408408```
409409
410410such that we have
411411
412412``` @raw html
413- <img src="img/diagram-braiding2.svg" alt="braiding relations" class="color-invertible"/>
413+ <img src="../ img/diagram-braiding2.svg" alt="braiding relations" class="color-invertible"/>
414414```
415415
416416where the expression on the right hand side, `` τ_{W,V}∘τ_{V,W} `` can generically not be
@@ -437,7 +437,7 @@ The braiding of a space and a dual space also follows naturally, it is given by
437437`` 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.
438438
439439``` @raw html
440- <img src="img/diagram-braidingdual.svg" alt="braiding dual" class="color-invertible"/>
440+ <img src="../ img/diagram-braidingdual.svg" alt="braiding dual" class="color-invertible"/>
441441```
442442
443443** Balanced categories** `` C `` are braided categories that come with a ** twist** `` θ `` , a
@@ -460,7 +460,7 @@ where we omitted the necessary left and right unitors and associators. Graphical
460460twists and their inverse (for which we refer to [ ^ turaev ] ) are then represented as
461461
462462``` @raw html
463- <img src="img/diagram-twists.svg" alt="twists" class="color-invertible"/>
463+ <img src="../ img/diagram-twists.svg" alt="twists" class="color-invertible"/>
464464```
465465
466466The graphical representation also makes it straightforward to verify that
@@ -486,7 +486,7 @@ structure, or, to define the exact pairing for the right dual functor as
486486or graphically
487487
488488``` @raw html
489- <img src="img/diagram-pivotalfromtwist.svg" alt="pivotal from twist" class="color-invertible"/>
489+ <img src="../ img/diagram-pivotalfromtwist.svg" alt="pivotal from twist" class="color-invertible"/>
490490```
491491
492492where we have drawn `` θ `` as `` θ^{\mathrm{l}} `` on the left and as `` θ^{\mathrm{r}} `` on
@@ -543,7 +543,7 @@ the morphism around a horizontal axis, and then reversing all arrows (bringing t
543543their original orientation before the mirror operation):
544544
545545``` @raw html
546- <img src="img/diagram-dagger.svg" alt="dagger" class="color-invertible"/>
546+ <img src="../ img/diagram-dagger.svg" alt="dagger" class="color-invertible"/>
547547```
548548
549549where for completeness we have also depicted the graphical representation of the transpose,
@@ -729,7 +729,7 @@ fusion category, on which we now focus, the corresponding projection maps are
729729Graphically, we represent these relations as
730730
731731``` @raw html
732- <img src="img/diagram-fusion.svg" alt="fusion" class="color-invertible"/>
732+ <img src="../ img/diagram-fusion.svg" alt="fusion" class="color-invertible"/>
733733```
734734
735735and also refer to the inclusion and projection maps as splitting and fusion tensor,
@@ -771,7 +771,7 @@ thus represent a unitary basis transform between the basis of inclusion maps
771771i.e. graphically:
772772
773773``` @raw html
774- <img src="img/diagram-Fmove.svg" alt="Fmove" class="color-invertible"/>
774+ <img src="../ img/diagram-Fmove.svg" alt="Fmove" class="color-invertible"/>
775775```
776776
777777The matrix `` F^{abc}_d `` is thus a unitary matrix. The pentagon coherence equation can also
@@ -786,7 +786,7 @@ triangle equation and its collaries imply that
786786`` F^{ab1}_c `` , which are graphically represented as
787787
788788``` @raw html
789- <img src="img/diagram-Fmove1.svg" alt="Fmove1" class="color-invertible"/>
789+ <img src="../ img/diagram-Fmove1.svg" alt="Fmove1" class="color-invertible"/>
790790```
791791
792792In the case of group representations, i.e. the category `` \mathbf{Rep}_{\mathsf{G}} `` , the
@@ -834,7 +834,7 @@ or thus ``χ_a = ±1``. This value is a topological invariant known as the
834834* Frobenius-Schur indicator* . Graphically, we represent this isomorphism and its relations as
835835
836836``` @raw html
837- <img src="img/diagram-Zisomorphism.svg" alt="Zisomorphism" class="color-invertible"/>
837+ <img src="../ img/diagram-Zisomorphism.svg" alt="Zisomorphism" class="color-invertible"/>
838838```
839839
840840We can now discuss the relation between the exact pairing and the fusion and splitting
@@ -855,14 +855,14 @@ encoded in the F-symbol. Hence, they do not represent new independent data. Agai
855855graphical representation is more enlightning:
856856
857857``` @raw html
858- <img src="img/diagram-ZtoF.svg" alt="ZtoF" class="color-invertible"/>
858+ <img src="../ img/diagram-ZtoF.svg" alt="ZtoF" class="color-invertible"/>
859859```
860860
861861With these definitions, we can now also evaluate the action of the evaluation map on the
862862splitting tensors, namely
863863
864864``` @raw html
865- <img src="img/diagram-splittingfusionrelation.svg" alt="splittingfusionrelation" class="color-invertible"/>
865+ <img src="../ img/diagram-splittingfusionrelation.svg" alt="splittingfusionrelation" class="color-invertible"/>
866866```
867867
868868where again bar denotes complex conjugation in the second line, and we introduced two new
@@ -880,7 +880,7 @@ the resulting element ``f ∈ \mathrm{End}(a)`` must satisfy
880880`` f = d_a^{-1} \mathrm{tr}(f) \mathrm{id}_a `` , i.e.
881881
882882``` @raw html
883- <img src="img/diagram-Brelation.svg" alt="Brelation" class="color-invertible"/>
883+ <img src="../ img/diagram-Brelation.svg" alt="Brelation" class="color-invertible"/>
884884```
885885
886886allows to conclude that
@@ -932,7 +932,7 @@ the simple objects. We can then express ``τ_{a,b}`` in terms of its matrix elem
932932or graphically
933933
934934``` @raw html
935- <img src="img/diagram-braidingR.svg" alt="braidingR" class="color-invertible"/>
935+ <img src="../ img/diagram-braidingR.svg" alt="braidingR" class="color-invertible"/>
936936```
937937
938938The hexagon coherence axiom for the braiding and the associator can then be reexpressed in
@@ -946,7 +946,7 @@ complex phases because of unitarity) multiplying the identity morphism, i.e.
946946or graphically
947947
948948``` @raw html
949- <img src="img/diagram-simpletwist.svg" alt="simpletwist" class="color-invertible"/>
949+ <img src="../ img/diagram-simpletwist.svg" alt="simpletwist" class="color-invertible"/>
950950```
951951
952952Henceforth, we reserve `` θ_a `` for the scalar value itself. Note that `` θ_a = θ_{\bar{a}} ``
@@ -959,7 +959,7 @@ If ``a = \bar{a}``, we can furthermore relate the twist, the braiding and the Fr
959959Schur indicator via `` θ_a χ_a R^{aa}_1 =1 `` , because of
960960
961961``` @raw html
962- <img src="img/diagram-twistfrobeniusschur.svg" alt="twistfrobeniusschur" class="color-invertible"/>
962+ <img src="../ img/diagram-twistfrobeniusschur.svg" alt="twistfrobeniusschur" class="color-invertible"/>
963963```
964964
965965For the recurring example of `` \mathbf{Rep}_{\mathsf{G}} `` , the braiding acts simply as the
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