The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

Real numbers are usually represented by various discrete objects such as floating points or partial decimal expansions. This is mainly because the clas- sical computability theory relates to computers which work with discrete data. Nevertheless, for theoretical purposes it is interesting to look at models of com- putation that deal with real numbers as with objects of unit size. A very natural such model was suggested by Blum, Shub and Smale in 1989. In 2012 Grigoriev and Nikolenko studied various cryptographic tasks involv- ing real numbers (for example, biometric authentication) and they considered the BSS machine model. In this work we focus on hard to invert functions in this model of computation. Our main theme is to analyse whether there are real functions of one variable that are easier to compute than to invert by a BSS machine. 1