The fourth and last paper in my series on ’The topology of critical processes’ is published, in Cahiers (as the previous parts): The topology of critical processes, IV (The homotopy structure) M. Grandis Abstract, Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover 'critical processes', indecomposable and un-stoppable - from the change of state in a memory cell to the action of a thermostat. The previous parts of this series introduced controlled spaces, examining how they can model critical processes in various domains, and studied their fundamental category. Here we deal with their formal homotopy theory. Cah. Topol. Géom. Différ. Catég. 66 (2025), no. 2, 46-93. https://cahierstgdc.com/index.php/volume-lxvi-2025/ ----- Part I: The topology of critical processes, I (Processes and Models) Cah. Topol. Géom. Différ. Catég. 65 (2024), no. 1, 3–34. https://cahierstgdc.com/index.php/volume-lxv-2024/ Part II: The topology of critical processes, II (The fundamental category) Cah. Topol. Géom. Différ. Catég. 65 (2024), no. 4, 438–483. https://cahierstgdc.com/index.php/volume-lxv-2024/ Part III: The topology of critical processes, III (Computing homotopy) Cah. Topol. Géom. Différ. Catég. 66 (2025), no. 1, 3–18. https://cahierstgdc.com/index.php/volume-lxvi-2025/ Marco Grandis e-mail: grandismrc@gmail.com You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>