## SCL Seminar by Branko Nikolic, 1st part

SCL seminar of the Center for the Study of Complex Systems, will be held on Wednesday, 17 January 2018 at 14:00 in the “Zvonko Marić” lecture hall of the Institute of Physics Belgrade. The talk entitled

will be given by Branko Nikolić (Department of Mathematics, Faculty of Science and Engineering, Macquarie University, Australia).

In this talk we will give a brief introduction into categories, which abstract mathematical objects and homomorphisms between them in a way similar to how groups abstract symmetry transformations or how numbers abstract counting [1]. Basic examples include sets (with functions), and vector spaces (with linear maps). Category is a common generalization of a group (more generally monoid) and an ordered set (more generally preorder). Similarly to group homomorphisms and order preserving maps, there are "category homomorphisms" called functors, also "functor homomorphisms" called natural transformations. Construction of a product of two categories enables a concise formal definition of a monoidal category. Monoidal categories are in turn a general framework for capturing (quantum) systems and processes, and representing them diagrammatically [2]. The definition of a category can be recast to use sets, functions and products of sets without referring to elements, enabling a generalization to categories enriched in an arbitrary monoidal category [3].

[1] S. Mac Lane, Categories for the Working Mathematician. Graduate Texts in Mathematics, Springer (New York, 1998).

[2] B. Coecke, Quantum Picturalism. Contemp. Phys. 51, 59 (2010).

[3] G. M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series vol. 64, Cambridge University Press (Cambridge-New York, 1982).

**"****Introduction to basic concepts of (enriched) category theory"**will be given by Branko Nikolić (Department of Mathematics, Faculty of Science and Engineering, Macquarie University, Australia).

**Abstract of the talk:**In this talk we will give a brief introduction into categories, which abstract mathematical objects and homomorphisms between them in a way similar to how groups abstract symmetry transformations or how numbers abstract counting [1]. Basic examples include sets (with functions), and vector spaces (with linear maps). Category is a common generalization of a group (more generally monoid) and an ordered set (more generally preorder). Similarly to group homomorphisms and order preserving maps, there are "category homomorphisms" called functors, also "functor homomorphisms" called natural transformations. Construction of a product of two categories enables a concise formal definition of a monoidal category. Monoidal categories are in turn a general framework for capturing (quantum) systems and processes, and representing them diagrammatically [2]. The definition of a category can be recast to use sets, functions and products of sets without referring to elements, enabling a generalization to categories enriched in an arbitrary monoidal category [3].

[1] S. Mac Lane, Categories for the Working Mathematician. Graduate Texts in Mathematics, Springer (New York, 1998).

[2] B. Coecke, Quantum Picturalism. Contemp. Phys. 51, 59 (2010).

[3] G. M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series vol. 64, Cambridge University Press (Cambridge-New York, 1982).