## Final Report Summary - PREDTOPOI (Predicative theories and Grothendieck toposes)

The project 'Predicative theories and Grothendieck toposes' had two main objectives: (1) training the fellow, Dr M Benini, assistant professor in computer science, Università degli Studi dell'Insubria, in the subject of mathematical logic, to make him an expert mathematician, specialised in the interpretation of constructive logical systems inside category theory; (2) producing novel knowledge about predicative theories, their categorical models, and their computational content. The proposal of the fellowship outlined a strategy to meet these goals that was focused on exploiting the information contained in models of a logical theory inside Grothendieck toposes, suggesting that models of predicative theories should be 'special' since they must carry deep computational content.

In the initial phase of the fellowship, it became clear that the planned strategy was not viable: the standard models in toposes do not necessarily represent the computational content, even when the originating theory is predicative, and, furthermore, Grothendieck toposes’ theory is not suitable, in the present state, to analyse first-order theories, i.e. the natural super-class of theories containing predicative ones. So, the fellow turned to study a novel class of logical models based on a different interpretation of logical theories inside category theory, where the computational content is inherently preserved and explicitly represented.

Specifically, a new notion, 'logically distributive category', has been synthesised, providing a sound and complete framework to interpret any system which can be expressed as an effective set of axioms based on the language of first-order intuitionistic logic, hence covering also the whole of predicative theories. The models in this semantics have an intrinsic computational content, deriving from the Curry-Howard isomorphism, even when the theory is not predicative. Thus, the initial goal has not only been achieved, but the obtained result is more general with respect to the initial goal of the project. Moreover, a precise correspondence has been developed to relate the interpretation inside logically distributive categories with the other forms of categorical semantics, namely Heyting categories, elementary toposes, Kripke models, and, finally, Grothendieck toposes, thus completing the proposal along a different line.

Also, a specific analysis of predicative theories reveals that the initial intuition was correct: predicative theories not only have a computational content, but they provide also a well-founded computational meaning, that is, each proof can be directly interpreted as a computer program written in a functional language which is immediately executable without assuming the existence of supporting procedures, as it happens with impredicative theories. Thus, referring back to the original proposal, the technical research goals have been completely met, with the proviso that they do not refer anymore to Grothendieck toposes, but to a different framework, logically distributive categories, which is strictly related.

One aspect of the initial project was philosophical: it asked to what extent the computational content of a logical theory shapes the interpretation in toposes. Using the categorical background developed in the fellowship, this question has been approached in the context of philosophy of information, analysing the programming activity from many points of view. In this respect, the fellow published a number of contributions to conferences and journals as a co-author, in a strict cooperation with a member of the community of philosophers of information.

Hence, the second goal of the project, research, has been substantially met in every aspect, and what has been achieved is more general than initially planned.

About the first goal, training, Dr Benini started from an already skilled position, being an academic staff member of his university. In this respect, it was important for him to shift from computer science to mathematics, where his interests in logic and computation could be developed in the right environment. In the fellowship period, he studied, attended to seminars in the School of Mathematics in the University of Leeds, and participated to the scientific life of the hosting department. In this way, he has been able to obtain the research results already described. But, thanks to those results, he is now both a recognised member in the international community of constructive mathematicians, and of philosophers of information. Apart publications, this fact is witnessed by his participation to two research projects: an international researchers exchange IRSES schema by EU, where Benini acts as the research unit leader in Università degli Studi dell'Insubria — the project has successfully concluded the negotiation phase; a research proposal to the Templeton Foundation where Benini is coordinating the research unit in his home university — the proposal has passed the first selection phase. It is noteworthy that, in both cases, Dr Benini operates as a mathematical logician, confirming the value of the fellowship experience, and the quality of training he received in Leeds.

Hence, the first goal has been partially achieved, since the publication record of the technical results is not yet complete. This incompleteness is due to lack of time, as the initial part of the fellowship, before discovering that the strategy in the proposal had no hopes, consumed the time which was initially planned for publishing the strictly mathematical outcomes. Nevertheless, the side research efforts in the philosophy of information proved to be more productive than expected, as the publication record in this aspect of the project has surpassed what initially planned. Also, the positioning action of the fellow inside the reference communities has been carried out with complete success, and his network of contacts has been widely extended, as his participation to international research projects in a leading position witnesses.

It is planned that, in the near future, more publications about the results obtained during the fellowship will appear. And it is expected that Dr Benini will continue his cooperation with the School of Mathematics in Leeds, as well as with the other members of the European mathematical community he met during the fellowship, in the form of joint research efforts.

In the initial phase of the fellowship, it became clear that the planned strategy was not viable: the standard models in toposes do not necessarily represent the computational content, even when the originating theory is predicative, and, furthermore, Grothendieck toposes’ theory is not suitable, in the present state, to analyse first-order theories, i.e. the natural super-class of theories containing predicative ones. So, the fellow turned to study a novel class of logical models based on a different interpretation of logical theories inside category theory, where the computational content is inherently preserved and explicitly represented.

Specifically, a new notion, 'logically distributive category', has been synthesised, providing a sound and complete framework to interpret any system which can be expressed as an effective set of axioms based on the language of first-order intuitionistic logic, hence covering also the whole of predicative theories. The models in this semantics have an intrinsic computational content, deriving from the Curry-Howard isomorphism, even when the theory is not predicative. Thus, the initial goal has not only been achieved, but the obtained result is more general with respect to the initial goal of the project. Moreover, a precise correspondence has been developed to relate the interpretation inside logically distributive categories with the other forms of categorical semantics, namely Heyting categories, elementary toposes, Kripke models, and, finally, Grothendieck toposes, thus completing the proposal along a different line.

Also, a specific analysis of predicative theories reveals that the initial intuition was correct: predicative theories not only have a computational content, but they provide also a well-founded computational meaning, that is, each proof can be directly interpreted as a computer program written in a functional language which is immediately executable without assuming the existence of supporting procedures, as it happens with impredicative theories. Thus, referring back to the original proposal, the technical research goals have been completely met, with the proviso that they do not refer anymore to Grothendieck toposes, but to a different framework, logically distributive categories, which is strictly related.

One aspect of the initial project was philosophical: it asked to what extent the computational content of a logical theory shapes the interpretation in toposes. Using the categorical background developed in the fellowship, this question has been approached in the context of philosophy of information, analysing the programming activity from many points of view. In this respect, the fellow published a number of contributions to conferences and journals as a co-author, in a strict cooperation with a member of the community of philosophers of information.

Hence, the second goal of the project, research, has been substantially met in every aspect, and what has been achieved is more general than initially planned.

About the first goal, training, Dr Benini started from an already skilled position, being an academic staff member of his university. In this respect, it was important for him to shift from computer science to mathematics, where his interests in logic and computation could be developed in the right environment. In the fellowship period, he studied, attended to seminars in the School of Mathematics in the University of Leeds, and participated to the scientific life of the hosting department. In this way, he has been able to obtain the research results already described. But, thanks to those results, he is now both a recognised member in the international community of constructive mathematicians, and of philosophers of information. Apart publications, this fact is witnessed by his participation to two research projects: an international researchers exchange IRSES schema by EU, where Benini acts as the research unit leader in Università degli Studi dell'Insubria — the project has successfully concluded the negotiation phase; a research proposal to the Templeton Foundation where Benini is coordinating the research unit in his home university — the proposal has passed the first selection phase. It is noteworthy that, in both cases, Dr Benini operates as a mathematical logician, confirming the value of the fellowship experience, and the quality of training he received in Leeds.

Hence, the first goal has been partially achieved, since the publication record of the technical results is not yet complete. This incompleteness is due to lack of time, as the initial part of the fellowship, before discovering that the strategy in the proposal had no hopes, consumed the time which was initially planned for publishing the strictly mathematical outcomes. Nevertheless, the side research efforts in the philosophy of information proved to be more productive than expected, as the publication record in this aspect of the project has surpassed what initially planned. Also, the positioning action of the fellow inside the reference communities has been carried out with complete success, and his network of contacts has been widely extended, as his participation to international research projects in a leading position witnesses.

It is planned that, in the near future, more publications about the results obtained during the fellowship will appear. And it is expected that Dr Benini will continue his cooperation with the School of Mathematics in Leeds, as well as with the other members of the European mathematical community he met during the fellowship, in the form of joint research efforts.