The procedure is more or less the standard Tannakian reconstruction argument. The first thing you need is a "forgetful" fiber functor $F:C\to Vect$, then you consider $R=End(F)$ the natural endomorphisms of the functor $F$ and this is the object you then show has the algebraic structure you require. Then the theorem is that $C$ is a category of representations over $R$.

For fusion categories you get that a generalized fiber functor always exists. You can consider the algebra $A$ to be the endomorphism algebra of the direct sum of (the finitely many isomorphism classes of) the simple objects, that is $A=End(\bigoplus_{S_{i}}S_{i})$ where $S_{i}$ are the (isomorphism classes) of the simple objects.

And then you can construct a functor $F:C\to A-Bimod$, this is simply equivalent to $Hom_{C}(\bigoplus S_{I},\_)$. Then it was shown that in fact $End(F)$ has a weak Hopf algebra structure and $C$ is a category of representations over it.

The proof that this has a weak Hopf algebra structure was originally (I think!) in:

Szlachányi, Kornél, Finite quantum groupoids and inclusions of finite type, Longo, Roberto (ed.), Mathematical physics in mathematics and physics. Quantum and operator algebraic aspects. Proceedings of a conference, Siena, Italy, June 20-24, 2000. Dedicated to Sergio Doplicher and John E. Roberts on the occasion of their 60th birthday. Providence, RI: AMS, American Mathematical Society. Fields Inst. Commun. 30, 393-407 (2001). ZBL1022.18007.

You can find this as Proposition/Exercise 7.23.11 and 7.23.12 of EGNO and around Remark 2.21 of:

Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor, On fusion categories., Ann. Math. (2) 162, No. 2, 581-642 (2005). ZBL1125.16025.

I hope this is of help, I am not at all familiar with the physics side of things so I cannot write this in the dialect that perhaps physicist uses so my apologies if this is not entirely clear.

The procedure is more or less the standard Tannakian reconstruction argument. The first thing you need is a "forgetful" fiber functor $F:C\to Vect$, then you consider $R=End(F)$ the natural endomorphisms of the functor $F$ and this is the object you then show has the algebraic structure you require. Then the theorem is that $C$ is a category of representations over $R$.

For fusion categories you get that a generalized fiber functor always exists. You can consider the algebra $A$ to be the endomorphism algebra of the direct sum of (the finitely many isomorphism classes of) the simple objects, that is $A=End(\bigoplus_{S_{i}}S_{i})$ where $S_{i}$ are the (isomorphism classes of the) simple objects.

And then you can construct a functor $F:C\to A-Bimod$, this is simply equivalent to $Hom_{C}(\bigoplus S_{i},\_)$. Then it was shown that in fact $End(F)$ has a weak Hopf algebra structure and $C$ is a category of representations over it.

The proof that this has a weak Hopf algebra structure was originally (I think!) in:

Szlachányi, Kornél, Finite quantum groupoids and inclusions of finite type, Longo, Roberto (ed.), Mathematical physics in mathematics and physics. Quantum and operator algebraic aspects. Proceedings of a conference, Siena, Italy, June 20-24, 2000. Dedicated to Sergio Doplicher and John E. Roberts on the occasion of their 60th birthday. Providence, RI: AMS, American Mathematical Society. Fields Inst. Commun. 30, 393-407 (2001). ZBL1022.18007.

You can find this as Proposition/Exercise 7.23.11 and 7.23.12 of EGNO and around Remark 2.21 of:

Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor, On fusion categories., Ann. Math. (2) 162, No. 2, 581-642 (2005). ZBL1125.16025.

I hope this is of help, I am not at all familiar with the physics side of things so I cannot write this in the dialect that perhaps physicist uses so my apologies if this is not entirely clear.

The procedure is more or less the standard Tannakian reconstruction argument. The first thing you need is a "forgetful" fiber functor $F:C\to Vect$, then you consider $R=End(F)$ the natural endomorphisms of the functor $F$ and this is the object you then show has the algebraic structure you require. Then the theorem is that $C$ is a category of representations over $R$.

For fusion categories you get that a generalized fiber functor always exists. You can consider the algebra $A$ to be the endomorphism algebra of the direct sum of (the finitely many isomorphism classes of) the simple objects, that is $A=End(\bigoplus_{S_{i}}S_{i})$ where $S_{i}$ are the (isomorphism classes) of the simple objects.

And then you can construct a functor $F:C\to A-Bimod$, this is simply equivalent to $Hom_{C}(A,\_)$. Then it was shown that in fact $End(F)$ has a weak Hopf algebra structure and $C$ is a category of representations over it.

The proof that this has a weak Hopf algebra structure was originally (I think!) in:

Szlachányi, Kornél, Finite quantum groupoids and inclusions of finite type, Longo, Roberto (ed.), Mathematical physics in mathematics and physics. Quantum and operator algebraic aspects. Proceedings of a conference, Siena, Italy, June 20-24, 2000. Dedicated to Sergio Doplicher and John E. Roberts on the occasion of their 60th birthday. Providence, RI: AMS, American Mathematical Society. Fields Inst. Commun. 30, 393-407 (2001). ZBL1022.18007.

You can find this as Proposition/Exercise 7.23.11 and 7.23.12 of EGNO and around Remark 2.21 of:

Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor, On fusion categories., Ann. Math. (2) 162, No. 2, 581-642 (2005). ZBL1125.16025.

I hope this is of help, I am not at all familiar with the physics side of things so I cannot write this in the dialect that perhaps physicist uses so my apologies if this is not entirely clear.

The procedure is more or less the standard Tannakian reconstruction argument. The first thing you need is a "forgetful" fiber functor $F:C\to Vect$, then you consider $R=End(F)$ the natural endomorphisms of the functor $F$ and this is the object you then show has the algebraic structure you require. Then the theorem is that $C$ is a category of representations over $R$.

For fusion categories you get that a generalized fiber functor always exists. You can consider the algebra $A$ to be the endomorphism algebra of the direct sum of (the finitely many isomorphism classes of) the simple objects, that is $A=End(\bigoplus_{S_{i}}S_{i})$ where $S_{i}$ are the (isomorphism classes) of the simple objects.

And then you can construct a functor $F:C\to A-Bimod$, this is simply equivalent to $Hom_{C}(\bigoplus S_{I},\_)$. Then it was shown that in fact $End(F)$ has a weak Hopf algebra structure and $C$ is a category of representations over it.

The proof that this has a weak Hopf algebra structure was originally (I think!) in:

Szlachányi, Kornél, Finite quantum groupoids and inclusions of finite type, Longo, Roberto (ed.), Mathematical physics in mathematics and physics. Quantum and operator algebraic aspects. Proceedings of a conference, Siena, Italy, June 20-24, 2000. Dedicated to Sergio Doplicher and John E. Roberts on the occasion of their 60th birthday. Providence, RI: AMS, American Mathematical Society. Fields Inst. Commun. 30, 393-407 (2001). ZBL1022.18007.

You can find this as Proposition/Exercise 7.23.11 and 7.23.12 of EGNO and around Remark 2.21 of:

Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor, On fusion categories., Ann. Math. (2) 162, No. 2, 581-642 (2005). ZBL1125.16025.

I hope this is of help, I am not at all familiar with the physics side of things so I cannot write this in the dialect that perhaps physicist uses so my apologies if this is not entirely clear.