  The Fourth International Conference on Applied Category Theory took place at
the Computer Laboratory of the University of Cambridge on 12--16 July 2021. It
was a hybrid event, with physical attendees present in Cambridge and other
participants taking part online. All the talks were recorded and the videos
have been posted online, links to which can be found on the conference website
(https://www.cl.cam.ac.uk/events/act2021/).
  Continuing the trend in the previous meetings of ACT, the contributions to
ACT 2021 ranged from pure to applied and represented a great variety of
categorical techniques and application topics, including: graphical calculi;
lenses; differential categories; categorical probability theory; machine
learning; game theory; cybernetics; natural language semantics and processing;
cryptography; and finite model theory.
  This proceedings volume contains about half of the papers that were presented
as talks at ACT 2021. This selection is a reflection of the authors' choice as
to whether to publish their papers in this volume or elsewhere.
