L&S Cat Theory: Attempted solutions to "Conceptual Mathematics" by Lawvere and Schanuel

Our given shape is defined as \(A_2 = \boxed{\bullet \rightarrow \bullet \rightarrow \bullet}\text{.}\) We have collect of “three dots” which I’ll name \(D_3 = \{d_0,d_1,d_2\}\) with \(\forall i: s d_i = t d_i = d_i\) and “two arrows” \(\{a_0,a_1\}\) which I’ll name \(a_0\) such that \(s a_0 = d_0\) and \(t a_0 = d_1\text{,}\) and \(a_2\) such that \(s a_2 = d_1\) and \(t a_2 = d_2\text{.}\)

There are “3 choose 2 = 3” ways that two of our three dots could overlap. In the case where \(d_0 = d_1\text{,}\)it produces the shape \(\boxed{\circlearrowleft \bullet \rightarrow \bullet}\text{,}\) for \(d_1 = d_2\) it produces \(\boxed{\bullet \rightarrow \bullet \circlearrowright}\text{,}\) and for \(d_0 = d_2\) it produces the shape \(\boxed{\bullet \mathrel{\substack{\rightarrow \\ \leftarrow}} \bullet} \text{.}\) There’s precisely one way all three dots could overlap such that \(d_0 = d_1 = d_2\) which has shape \(\boxed{\circlearrowleft \bullet \circlearrowright}\text{.}\) This last option admits one additional way of being singular, and that is when \(a_0 = a_1\text{,}\) this produces the final shape \(\boxed{\bullet \circlearrowright}\text{.}\)

In Session 15, our endomaps \(X^{\circlearrowright \alpha}\) and \(Y^{\circlearrowright \beta}\) had the following shapes:

Since these are both endomaps, we know there’s a isomorphism betweeen dots and arrows. For every \(x_i \in X\) there’s a unique arrow \(a_i\) originating at the respective dot, and similarly an arrow \(b_j\) for each \(y_j \in Y\text{.}\) What varies from dot to dot are the number of arrows with that dot as a target. During Session 15 (Part 5) 4.14 I was referring to a map I called “incoming arrow count”, which I think might connected to this notion of “incidence”, but at the time I was focused more on dots than arrows so maybe something will reveal itself by looking at them as arrows instead.

I’m also wondering if this object \(A_2\) that was introduced here is related to what I’ve been calling “tails”. If I take \(A_0\) to be the “naked dot” and \(A_1\) to be the naked arrow, then I can use \(A_n\) to represent arbitrarily long distances to a terminal object. This might give me a more objective way of defining the map I previously called “number of steps to stabilize”.

Speaking of terminal objects, I’ve been thinking that my answer to Article 4 Exercise 4 might be incorrect. My assumption that \(Y = \mathbf{1}\) misses the fact that any n-cycle \(C_n\) is potentially a terminal object in \(\mathcal{S}^{\downarrow_\bullet^\bullet \downarrow}\text{.}\) I think the distinction that I failed to make was that \(s\) and \(t\) aren’t necessarily the same map, they just need to have inverses because there exactly one arrow leading into and out of each point. Perhaps the terminal object in \(\mathcal{S}^{\downarrow_\bullet^\bullet \downarrow}\) could be defined diectly as some factor of the identity map.

The other thing that I’ve been curious about was Article 4 Exercise 6. What if the special property of \(\mathbb{N}\) being hinted at here was really the uniqueness of prime factorizations? Given any number \(n\text{,}\) there’s a unique representation \(n = p_1^{q_1} \times p_2^{q_2} \times ... \times p_n^{q_n}\) where \(p_i\) are the prime numbers and \(q_j \in \mathbb{N}\text{.}\) This would explain our little side venture to prove \(C_24 \rightarrow C_8 \times C_3\) in Session 21 (Part 1) 5.12.

After playing around with this a bit in Python

¹

github.com/rruff82/LS-Categories/blob/main/assets/notebooks/ls-ses22.ipynb

, I have a couple of observations. First, I noticed that my “incoming arrow counts” is essentially “G.degree(N)-1” and I think this off-by-one error might be related to why I was short maps in Session 15.

The other thing I started thinking about is the choice of notation. I began wondering if the words choice of “arrow”, “binary”, “cycle”, and “dot” has a double meaning when interpreted as the sequence of maps \(A \rightarrow B \rightarrow C \rightarrow D\text{.}\) If \(I,T\) are the initial and terminal objects in the category, any map \(X \xrightarrow{f} Y\) corresponds to analagous sequence \(I \rightarrow X \xrightarrow{f} Y \rightarrow T\) by treating the naked arrow \(A\) as the initial object \(I\) and naked dot \(D\) as the terminal object \(T\text{.}\)

It also seemed weird how the authors introduce the “heavy arrow” to represent “epimorphism” then proceed to not use it again for the rest of the session. If the naming of objects \(A,B,C,D\) has a double meaning, then maybe the choice of \(r,s,t\) are important also. Perhaps the maps defined by “retraction”, “source”, and “target” can be defined by sasifying the associative property on the sequence of maps \(I \xrightarrow{r} X \xrightarrow{s} Y \xrightarrow{t} T\text{.}\)

I also found myself finding interesting properties of the maps \(\mathbb{N} \rightleftarrows \mathbb{N} \times \mathbb{N}\text{.}\) I think I’m starting to understand the relevance of defining incidence relations. By “stepping out” through mapping \(\langle s,t \rangle \rightarrow n_1\) and \(\langle t,s \rangle \rightarrow n_2\) we can use the unique \(\mathcal{S}\)-map \(\{n_1\} \times \{n_2\} \rightarrow \{n\}\) compare whether or not \(s\) and \(t\) are the “same map”. If they are, then we have a terminal object in \(\mathcal{S}^{\bullet \downarrow_\bullet^\bullet \downarrow}\text{.}\) If they are not, then define have a new pair of maps \(s',t'\) which we can apply the process two a second time.