# Rat between two walls viewed as a product in a category

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.

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!

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

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.