Dr. Branko Nikolić (Department of Mathematics, Faculty of Science and Engineering, Macquarie University, Australia) was our guest at SCL seminar of the Center for the Study of Complex Systems on Wednesday, 17 January and on Thursday, 18 January 2018. He held two talks entitled:

“Introduction to basic concepts of (enriched) category theory”

“Metric and (relativistic) event spaces via enrichment”

**Abstract of the talk “Introduction to basic concepts of (enriched) category theory”:**

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).

**Abstract of the talk “Metric and (relativistic) event spaces via enrichment”:**

When the base of enrichment is a monoidal category that is in fact a poset (aka quantale), the definition of an enriched category becomes simpler. In this talk, we will consider two particular quantales consisting of positive real numbers. Categories enriched in the first are generalized metric spaces [1]. The categories enriched in the second can be interpreted as causal preorders that remember intervals (times) between time-like events [2]. Modules between enriched categories enable expressing Cauchy completeness of metric spaces in purely categorical terms; in this sense all event spaces are Cauchy complete. We will give sufficient conditions on a monoidal category that ensure that an enriched category is Cauchy complete if and only if idempotents split in its underlying category.

[1] F. W. Lawvere, Metric Spaces, Generalized Logic, and Closed Categories, Seminario Mat. e. Fis. di Milano 43, 135 (1973).

[2] B. Nikolic, Cauchy Completeness and Causal Spaces, arXiv:1712.00560v1 [math.CT] (2017).