### Lecture Videos

Session 5 Chapter 3: Databases: Categories, Functors, and (Co)Limits Part 1 by Dr. David I. Spivak |

Session 6 Chapter 3: Databases: Categories, Functors, and (Co)Limits Part 2 by Dr. Brendan Fong |

### Readings

Internal links in the chapter file below are non-functional. For working links, open the file for the full textbook: *An Invitation to Applied Category Theory: Seven Sketches in Compositionality* (PDF - 2.6MB).

Chapter 3: Databases: Categories, Functors, and (Co)Limits (PDF)

3.1 What is a database?

3.2 Categories

3.2.1 Free categories

3.2.2 Presenting categories via path equations

3.2.3 Preorders and free categories: two ends of a spectrum

3.2.4 Important categories in mathematics

3.2.5 Isomorphisms in a category

3.3 Functors, natural transformations, and databases

3.3.1 Sets and functions as databases

3.3.2 Functors

3.3.3 Database instances as **Set**-valued functors

3.3.4 Natural transformations

3.3.5 The category of instances on a schema

3.4 Adjunctions and data migration

3.4.1 Pulling back data along a functor

3.4.2 Adjunctions

3.4.3 Left and right pushforward functors, \(\Sigma\) and \(\Pi\)

3.4.4 Single set summaries of databases

3.5 Bonus: An introduction to limits and colimits

3.5.1 Terminal objects and products

3.5.2 Limits

3.5.3 Finite limits in **Set**

3.5.4 A brief note on colimits

3.6 Summary and further reading

### Assignment

Problem set 2 (PDF) due at the beginning of Session 10.