I'm reading the book "A New Foundation for Representation in Cognitive Brain Science - Category Theory and the Hippocampus" by Jaime Gómez-Ramirez (a terrible book, if you ask me).

On p. 164 the author introduces the product of two objects in a category. I.e. the product of two objects $A$ and $B$ in a category $\mathcal C$ is another object $P$ equipped with two morphisms $p_1: P\rightarrow A$ and $p_2: P\rightarrow B$ such that for any pair of morphisms $x_1: X\rightarrow A$ and $x_2: X\rightarrow B$ there is a unique morphism $h$ making the figure commute.

enter image description here

The author goes on to say that the main characteristic of a product is that the constituents are retrievable via the projections. The following example is given: Let $W_A$ and $W_B$ be two walls that the delimit the maze a rat is moving in. After reaching both walls, the rat would develop the concept of a middle point $P$. This middle point $P$ corresponds to the categorial product $P=W_A \times W_B$.

I struggle to make sense of this. If the walls, or more generally, any location in the maze, are objects in a category, then what are the arrows (morphisms)? My first guess is: paths. Translated into the maze-world: There is a "concept" $W_A \times W_B$ that gives me paths from each wall to the middle $P$. And given any point $X$ in the maze there is a unique path $X\rightarrow P$ from $X$ to $P$. This interpretation doesn't make sense at all.

The second guess is that the walls $W_A, W_B$ are not actual locations in space, but the rat's concepts of these walls. But it is still unclear, what the arrows between concepts are supposed to be.

Is there a way to make sense of this? Specifically, what exactly is this "maze-category"? Because to me, so far, this is just half-baked, abstract nonsense jibber jabber.

Any pointers to more substantial literature would be greatly appreciated!

  • $\begingroup$ I don't have any knowledge on this but I wondered if the Wikipedia article on Category Theory might help you $\endgroup$ – Chris Rogers Mar 27 '18 at 11:27
  • $\begingroup$ Category theory as such is not my problem. It is the application thereof in this context. $\endgroup$ – mcmayer Mar 27 '18 at 12:20

According to the author, in this maze-category the objects are memories (or mental objects).

The author references to chapter 3, when discussing the maze-category. Chapter 3 of the book, gives an application to Memory Evolutive Neuronal Systems (MENS). There, a morphism represents a way of relating mental objects. This is a general description.

The author gives as examples for the maze-category:

  • travel time
  • number of steps needed to go from one mental object to another.

The product $P$ of two memories is two memories at once. The concept of a middle point requires that the agent has the memories of both memories. One wall only requires one memory.

$X$ can potentially represent any of the points between the two walls.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.