Constructing condensed memories in functorial time

ABSTRACT

If episodic memory is constructive, experienced time is also a construct. We develop an event-based formalism that replaces the traditional objective, agent-independent notion of time with a constructive, agent-dependent notion of time. We show how to make this agent-dependent time entropic and hence well-defined. We use sheaf-theoretic techniques to render agent-dependent time functorial and to construct episodic memories as sequences of observed and constructed events with well-defined limits that maximise the consistency of categorisations assigned to objects appearing in memories. We then develop a condensed formalism that represents episodic memories as pure constructs from single events. We formulate an empirical hypothesis that human episodic memory implements a particular time-symmetric constructive functor, and discuss possible experimental tests.

KEYWORDS:

• condensed sets
• pro-objects
• frame problem
• hypergraphs
• sheaves

Disclosure statement

No potential conflict of interest was reported by the author(s).

Author contributions

SD and CF jointly developed the results and wrote the paper.

Notes

1. A category comprises a set of objects and a set of morphisms between objects, subject to the requirements that 1) every object has an associated Identity morphism and 2) morphisms compose associatively. A functor is a mapping between categories that respects Identity morphisms and morphism composition. Finite sets with functions, and groups, rings, fields, or topological spaces with their associated homomorphisms all form categories; indeed all of mathematics can be formulated in category-theoretic terms. For introductions to the theory, see e.g. Adámek, Herrlich and Strecker (1990); https://en.wikipedia.org/wiki/Category_(mathematics), or the extensive resources of https://ncatlab.org/.

2. We are assuming here that it is meaningful to talk about individuals being identifiable as such over time, an assumption that is questioned by multiple philosophical traditions. We do not engage here with this philosophical debate, but rather explore, in what follows, how observers go about identifying individuals, and how this process relates to the subjective experience of time.

3. Again see Fields & Glazebrook (2019a) for discussion and examples. Channel theory has been reformulated in the alternative language of institutions (Goguen & Burstall, 1992) in Goguen (2004).

4. All standard definitions not otherwise referenced are from https://ncatlab.org/nlab/show/HomePage.

5. As we began in 2 with an informal notion of the relation between experienced events, we can simply stipulate that their topology is Grothendieck. Nothing we have done is inconsistent with this slightly stronger notion of discreteness.