1st semester 
20 credits / 30 lectures

Category Theory is a kind of ‘third level’ of mathematics after the classical and set-theoretic ones. It develops general constructions, theorems and arguments, applicable to a wide range of diverse types of mathematical structures, which allows unifying and simplifying many mathematical theories, including the very foundation of mathematics. In some sense, every word of category theory is just about composing arrows between points, and yet this seemingly so poor language is powerful enough to express everything set theory can express, and often to lead to great clarifications and new results. This course is about the first basic notions of category theory, about their main properties, and about above-mentioned applications and clarifications in other areas of pure mathematics. The list of topics will include:

• Categories, functors, natural transformations
• Equivalence of categories
• Yoneda lemma and Yoneda embedding
• Universal properties and representability
• Limits and colimits
• Adjoint functors
• Higher categorical structures

with many examples everywhere, especially in algebra, topology, and mathematical logic.