## SCL Seminar by Branko Nikolic, 2nd part

SCL seminar of the Center for the Study of Complex Systems, will be held on Thursday, 18 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).

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).

**"****Metric and (relativistic) event spaces via enrichment"**will be given by Branko Nikolić (Department of Mathematics, Faculty of Science and Engineering, Macquarie University, Australia).

**Abstract of the talk:**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).