Improve SSiC#

from IPython.display import Image, Markdown

Optimization problems#

See How are universal properties “solutions to optimization problems”?. Example 3.99 (introducing the pullback) seems like a good example of this; we have to construct some object based on constraints the object must satisfy (it literally uses the word “constraint”). See “equality constraints” in Optimization problem. It would be helpful to add certain “hard” equality constraints to the “soft” objective function minimization/maximization that is often done in an optimization.

See also Equaliser (mathematics), and the database constraints in Chp. 3.

Isomorphic objects#

Why do we call two items isomorphic in a pre-order? Because in one shot (one arrow) you can go to an “equivalent” node on the other side. You could see these two nodes as two one-object categories that are equivalent. To generalize this notion of “isomorphic” to two object categories, you just need two arrows that are bidirectional.

Recursion and terminal objects#

Recursive base case for extended real number line is infinity. Serves as terminal object. Terminate program.

6.2.1 Initial objects#

Russell’s analysis in forallx looks related to unique objects; see Chapter 28.

Exercise 6.7²#

Part 2.#


For part 2. can we use a one-element set? See Field with one element; at least for a field this isn’t a well-defined concept. However, why can’t the multiplicative identity equal the Additive identity? See Additive identity § Properties; there’s no hard reason this isn’t possible for a ring. In fact, this is known as the Zero ring.

However, the zero ring cannot serve as an initial object in Rig because we can still preserve structure and map the one element in this structure to either the additive or multiplicative identity in many other rigs (such as e.g. the natural numbers). That makes two morphisms (homomorphisms) from the zero ring to the natural numbers, when an initial object must only have one.

Having both a multiplicative and additive element implies (via the distributive property) that some other element 1+1 must exist, and via mathematical induction this implies we must have at least the natural numbers in an initial object. Are the natural numbers an initial object in Rig? There is exactly one morphism from them to the zero ring (all elements go to zero).

Without reading it, this solution looks long. For likely solutions to this question (check before checking the back of the book), see:

Interpretations of zero#

The symbol 0 gets used in higher mathematics for much more than the number zero; see Zero element. In part (d) of Definition 5.36 (a dependency of Exercise 6.7) it’s specifically identified as an absorbing element, besides being the additive identity. Since we’re in a section on initial objects, and have already learned about terminal objects, it may be worth looking at the concept of a Zero object (algebra) (which is both initial and terminal).

All over the article Zero object (algebra), the word “trivial” is used almost as a synonym for zero object. At the start:

This article is about trivial or zero algebraic structures.

The article Initial and terminal objects redefines “zero object” but links to Zero object (algebra) and specifically mentions:

This is the origin of the term “zero object”.

In Initial and terminal objects, the symbol 0 is often used for initial objects (though the author uses ∅). It’s likely this is because 0 serves as the additive identity and so seems natural alongside the symbol + often used for the coproduct. The symbol 1 is similarly often used for terminal objects because it is the identity with respect to the categorical product. The article also mentions:

Cat, the category of small categories with functors as morphisms has the empty category, 0 (with no objects and no morphisms), as initial object and the terminal category, 1 (with a single object with a single identity morphism), as terminal object.

While it often works to use 0 for an initial object and 1 for a terminal object, this also not a specific approach. It can also be confusing when a category has a “zero” object that is not initial (as in Ring/Rig, where the zero ring/rig is terminal).