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Stream: deprecated: public research

Topic: Affine categories

Jean-Baptiste Vienney (Jul 24 2022 at 16:06):

I've started to work on this idea of affine category which is a generalization of affine spaces. If somebody is interested by this idea, he could find fun to work on it.

Jean-Baptiste Vienney (Jul 24 2022 at 16:07):

What is the idea?

Jean-Baptiste Vienney (Jul 24 2022 at 16:08):

Let’s think to how a space can be described as a category.

Jean-Baptiste Vienney (Jul 24 2022 at 16:09):

For a topological space, it is something used in algebraic topology.

Jean-Baptiste Vienney (Jul 24 2022 at 16:12):

If $X$ is a topological space, you obtain a category $\mathcal{C}$ by saying that:

• The set of objects is $X$
• For every $x,y \in X$, $\mathcal{C}[x,y]$ is the set of paths from $x$ to $y$ ie. the continuous applications $f:[0,1] \rightarrow X$ such that $f(0)=x$ and $f(1)=y$

Jean-Baptiste Vienney (Jul 24 2022 at 16:12):

The identity and composition are the natural ones.

Jean-Baptiste Vienney (Jul 24 2022 at 16:13):

An affine space can be described somewhat similarly as a category.

Jean-Baptiste Vienney (Jul 24 2022 at 16:15):

It will be the simplest example of what we could call an affine category.

Jean-Baptiste Vienney (Jul 24 2022 at 16:15):

We must make clear our definition of a category because in an affine category, it is very important that the different hom-sets are not supposed disjoint. Thus, we must be sure that composing two morphisms by the intermediate of different hom-sets give the same result.

Jean-Baptiste Vienney (Jul 24 2022 at 16:17):

Definition:
A category $\mathcal{C}$ is given by a set $Ob_{\mathcal{C}}$, for every $a,b \in Ob_{\mathcal{C}}$, a set $\mathcal{C}[a,b]$, for every object $a \in Ob_{\mathcal{C}}$, an element $1_{a} \in \mathcal{C}[a,a]$ and for every $a,b,c \in Ob_{\mathcal{C}}$, a function
$\mathcal{C}[a,b] \times \mathcal{C}[b,c] \rightarrow \mathcal{C}[a,c]$ which associates $f;g$ to $(f,g)$
such that the composition is consistent, in the sense that if $f,g$ can be composed by using different hom-sets, we obtain the same composite in every case, and such that $(f;g);h = f;(g;h)$ and $f;1_{b} = 1_{a};f = f$ if $f \in \mathcal{C}[a,b]$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:20):

You can go to Affine space for the usual definition of an affine space.

Jean-Baptiste Vienney (Jul 24 2022 at 16:21):

I will give a more categorical but equivalent definition. (This is an example of the notion of affine category which is defined below.)

Jean-Baptiste Vienney (Jul 24 2022 at 16:25):

Example:
An affine space over a field $\mathbb{K}$ is a category $\mathcal{C}$ such that for every object $a,b \in Ob_{\mathcal{C}}$, $\mathcal{C}[a,b]$ is a singleton (we identify this singleton with its only element), $Mor_{\mathcal{C}}$ is a vector space over $\mathbb{K}$ and for every object $a \in Ob_{\mathcal{C}}$, the function $\mathcal{C}[a,-]: Ob_{\mathcal{C}} \rightarrow Mor_{\mathcal{C}}$ which associates $\mathcal{C}[a,b]$ to $b$ is a bijection. We also require that $f;g = f+g$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:29):

The idea of an affine space is to describe vectors between points in a space.

Jean-Baptiste Vienney (Jul 24 2022 at 16:29):

The objects of our category must be considered as the points of a space.

Jean-Baptiste Vienney (Jul 24 2022 at 16:30):

$\mathcal{C}[a,b]$ is a singleton which is constitued by the only vector $\overset{\rightarrow}{ab}$ in more usual terms.

Jean-Baptiste Vienney (Jul 24 2022 at 16:33):

The composition associates $\overset{\rightarrow}{ac} = \overset{\rightarrow}{ab} + \overset{\rightarrow}{bc}$ to $(\overset{\rightarrow}{ab},\overset{\rightarrow}{bc})$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:36):

The requirement of bijection ensures that for every point $a$, for every vector $v$, there is one and only one object $b$ such that $v=\overset{\rightarrow}{ab}$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:37):

Now, the definition of an affine category:

Jean-Baptiste Vienney (Jul 24 2022 at 16:37):

Definition:
An affine category is a category $\mathcal{C}$ such that for every objects $a,b \in \mathcal{C}$, $\underset{x \in \mathcal{C}}{\bigcup} \mathcal{C}[a,x] = \underset{x \in \mathcal{C}}{\bigcup} \mathcal{C}[b,x]$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:38):

Proposition:
A category is an affine category iff for every $a \in \mathcal{C}$, $Mor_{\mathcal{C}} = \underset{x \in \mathcal{C}}{\bigcup} \mathcal{C}[a,x]$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:39):

There is lots of affine categories which are funnier that an affine space. The idea is that now, the objects of the affine category are still the points of our space but there is way more possibilities for vectors between these points.

Jean-Baptiste Vienney (Jul 24 2022 at 16:42):

Proposition:
In an affine category, for every $a,b \in Ob_{\mathcal{C}}$, $1_{a}=1_{b}$. We note this object $0$. $Mor_{\mathcal{C}}$ is a monoid of unit $0$ by defining $f+g$ equal to the composite $f;g$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:42):

Now we can have different morphisms $a \rightarrow b$ not only one vector. These morphisms are thus more paths than vectors. In an affine space, there is only one such path which is given by $\overset{\rightarrow}{ab}$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:46):

However, mere topological spaces are not affine categories. Because there is an idea of uniformity in an affine category: the paths starting from any point $a$ are the same than the paths starting from any point $b$.

Jean-Baptiste Vienney (Jul 24 2022 at 16:48):

Example:
$\mathbb{R}^{n}$ is an affine category for every $n \ge 1$. A morphism $a \rightarrow b$ is given by an equivalence class $\overline{f}$ where $f$ is a path from $a$ to $b$ and under the equivalence relation $\sim$ on the set of all paths from any point to any point (considered as continuous functions $[0,1] \rightarrow \mathbb{R}^{n}$) given by $f \sim g$ iff $f-g$ is a constant. It means that a morphism $a \rightarrow b$ is a path from $a$ to $b$ but when you translate to the same path $a' \rightarrow b'$ it's always considered as the same morphism.

Jean-Baptiste Vienney (Jul 24 2022 at 16:58):

We can see with this example that in an affine category, $Mor_{\mathcal{C}}$ is not necessarily a commutative monoid such as in an affine space. When you compose two vectors in an affine space, the order doesn't matter because it is a very simple case.

Jean-Baptiste Vienney (Jul 24 2022 at 17:00):

There is also an example by considering the tops of a grid as the objects and the paths on this grid as morphims but it would be some work to draw it.

Also, a sphere of any dimension seems to be an example with morphisms defined as for $\mathbb{R}^{n}$.

Jean-Baptiste Vienney (Jul 24 2022 at 17:02):

If you’re interested by this project,

Jean-Baptiste Vienney (Jul 24 2022 at 18:11):

I would be happy to clarify any point or to learn from you! I'm sure someone who knows geometry would have good ideas! but pointing any unclear point would be as great.

Jean-Baptiste Vienney (Jul 24 2022 at 18:33):

If not, I would continue to have fun alone!