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

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## 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

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).

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

3.4.1 Pulling back data along a functor

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