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

<p>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. </p>

<p>The author gives as examples for the maze-category:</p>

<ul>
<li>travel time </li>
<li>number of steps needed to go from one mental object to another.</li>
</ul>

<p>The product <span class="math-container">$P$</span> 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.</p>

<p><span class="math-container">$X$</span> can potentially represent any of the points between the two walls. </p>

CC BY-SA 4.0
2020-05-07T19:16:05.630Z
2020-05-07T19:28:02.293Z
2020-05-07T19:28:02.293Z
Nick Decroos
<p>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).</p>

<p>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.</p>

<p><a href="https://i.sstatic.net/2lXpS.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2lXpS.jpg" alt="enter image description here"></a></p>

<p>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 $P=W_A \times W_B$.</p>

<p>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.</p>

<p>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.</p>

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

<p>Any pointers to more substantial literature would be greatly appreciated!</p>

CC BY-SA 3.0
2018-03-27T11:27:40.177Z
I don't have any knowledge on this but I wondered if the [Wikipedia article on Category Theory](https://en.wikipedia.org/wiki/Category_theory) might help you
Chris Rogers
CC BY-SA 3.0
2018-03-27T12:20:00.340Z
Category theory as such is not my problem. It is the application thereof in this context.
mcmayer
<cognitive-modeling><mathematical-psychology>
Rat between two walls viewed as a product in a category
mcmayer
# Rat between two walls viewed as a product in a category

(Asked by: username_0 on 2018-03-27)

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](https://i.sstatic.net/2lXpS.jpg)](https://i.sstatic.net/2lXpS.jpg)

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!

**Tags:** cognitive-modeling, mathematical-psychology

> **Comments:**
> **username_1**: I don't have any knowledge on this but I wondered if the [Wikipedia article on Category Theory](https://en.wikipedia.org/wiki/Category_theory) might help you  
> **username_0**: Category theory as such is not my problem. It is the application thereof in this context.  

---

## 1 answers

### Answer by username_2 on 2020-05-07. Score: 1

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.


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