Following is my attempt to construct a category and some questions:
Consider the set of integers $\Bbb{Z}$ as objects.
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There is a morphism $A \to B$ if there exists some number $c$ such that $A + c = B$. Henceforth denoted $A \xrightarrow{c} B$
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There is an identity where $c = 0$.
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Morphisms can be composed. $A \xrightarrow{a} B \xrightarrow{b} C = A \xrightarrow{a+b} C$.
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e.g. $1 \xrightarrow{1} 2 \xrightarrow{2} 4 = 1 \xrightarrow{3} 4$
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This composition is associative. The composition is just addition of differences, and addition is associative.
Ex. $1 \xrightarrow{1} 2 \xrightarrow{2} 4 \xrightarrow{3} 7 == 1 \xrightarrow{1 + (2 + 3)} 7 == 1 \xrightarrow{(1 + 2) + 3} 7$ -
Composition with the identity, $(c + 0) = c$, follows trivially from the definition of zero and addition.
From the above characteristics, it follows that
$A + c = B$ is a category.
Observation:
On $\Bbb{Z}$, there is a morphism between every ob and every other ob, since $c = B – A$ and $x – y$ is an integer for all integer input.
Consider the above defintion, but instead on $\Bbb{Z}^*$ (non-negative integers).
The defining characteristics of the category still hold true, but the observation does not.
There is no morphism from $2$ to $1$, as $c$ would need to be $-1$, not contained in $\Bbb{Z}^*$.
Specifically, there are morphisms where $A \leqslant B$.
Consider the same definition, but on $\Bbb{Z}^+$ (positive integers). This is the same set as $\Bbb{Z}^*$, but without zero.
Observe that this definition does not constitute a category. Specifically, there is no identity, as for a morphism $A \xrightarrow{c} A$ to exist, $c$ must equal zero.
Questions:
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Did I make any mistakes above?
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One source of confusion for me is the difference between this category and a monoid. The examples I described as categories are clearly monoids, and indeed non-negative integers with addition is wikipedia's example of a monoid. However, it adds this confusing (to me) line, saying "More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object".
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It seems to me that there's an infinite number of objects as I've defined my category. Is there some other category that wikipedia is talking about, or am I misunderstanding something in my definition of objects in this category?
Best Answer
A smaller example might be useful to address your questions. Let $M = \{1,a,b\}$ be the monoid with identity $1$ defined by $aa = ba = a$ and $bb = ab = b$. There are several ways to associate a category to $M$. The first way is similar to your construction. You take $M$ as set of objects and the morphisms are of the form $x \xrightarrow{r}y$ where $xr = y$. For $M$, you get the category pictured below:
$\hskip 100pt$
About the "confusing line" the morphisms of an object to itself form a monoid. The morphisms from $1$ to $1$ form the monoid $\{1\}$, the morphisms from $a$ to $a$ form the monoid $\{1, a\}$ and the morphisms from $b$ to $b$ form the monoid $\{1, b\}$.
A second way is to associate a category to $M$ is suggested in the second part of the "confusing line". It consists to take the one object category represented below:
$\hskip 160pt$