My aim is to understand how the notion of su-bobject classifier is used in the article Topos and Stacks of Deep Neural Networks about the categorical formalization of deep neural networks. This article has very recent addition in the form of chapter Mathematics for AI: Categories, Toposes, Types of the book that is being published 2022.02.
This chapter contains the following text:
and
So, I am trying to understand the following questions:
The authors talk about the category $C|X$ whose objects are the objects of the original category $C$. This original category is category of nodes and paths that is defined for one specific graph that describes the architecture of the deep neural network under consideration. The article defines the functor-presheaf for this category, functor that assigns set (or real numbers) for each node. Traditional interpretation is that this set contains only single element for each node, the activation value of the neuron for the node. So, actually each functor-presheaf is the concrete assignment of concrete numerical activations to the whole DNN. The set of all such functors forms the topos $C^{\wedge}$. And this is my confusion - the authors (furhter in the text) are trying to construct the subobject classifier for the topos $C^{\wedge}$ and not for the category $C$. So, maybe it would have been better if they defined here the category $C^{\wedge}|F$ (where $F$ is the functor-presheaf), because, my guess, they are talking exactly about this category (created from the topos $C^{\wedge}$ and not from the initial graph category $C$)?
For the one moment I thought that they are considering presheaf topos $(C|X)^{\wedge}$ of presheafs-functors that are functors from the morphisms in $C$ (i.e. arrows among nodes) to the the sets (again - one element set, that is assigned to each arrow) and hence that can be interpreted as the concrete assignment of the weights to the arrows in the DNN. And $\textbf{1}|X$ can be the functor (arrow-to-set) that assigns constant numerical value (one-element-sets that contain this number) to all the arrows (that ar going into node $X$).
But further discussion invalidates this hypothesis. It seems to me that $\textbf{1}|X$ is still the presheaf in the full topos $C^{\wedge}$.
And here we are - how to interpret the symbol $\Omega _X$ - is it an nobject in topos $C^{\wedge}$ or in the topos (suggested by me, this suggestion corrects the eventual presentation misunderstanding of this article) $(C|X)^{\wedge}$?
So, what is meant by $\Omega _X$ in this article? And how $\Omega _X$ is related to $\Omega$ and how this can be understood in the terms of neural network (nodes, arrows, assignment of activations, assignment of weights, mapping among the 2 assignments of activations, mapping among the 2 assignment of weights)?
$\Omega_X$ is the subobject classifier of $(C|X)^\vee$, the catgory of presheaves on $C|X$. $C|X$ itself is the category whose objects are paths to $X$ in the original category $C$, and whose morphisms from one path to another an initial path of the former, so that the latter is the remaining path.There is an equivalence of categories between $(C|X)^\vee$ and $C^\vee|X^\vee$. It arises because $C|X$ with its domain functor $C|X\to C$ is the result of the Grothendieck constructoin applied to the presheaf $X^\vee$. In detail, $\mathrm{dom}\colon C|X\to C$ is the discrete fibration corresponding to the presheaf $X^\vee$, and any presheaf on $C|X$ then corresponds to a discrete fibration $p\colon F\to C|X$, which is the same data as a morphism $p$ from a discrete fibration $\mathrm{dom}\circ p\colon F\to C$ to the discrete $\mathrm{dom}\colon C|X\to C$, corresponding to a natural transformation from a presheaf on $C$ to the presheaf $X^\vee$.
Explicitly, a presheaf $F$ on $C$ equipped with a natural transformation $FX\to X^\vee$ is a family of set-functions from $FY$ to the set of morphisms $Y\to X$, so that pre-composing with the morphism $Z\to Y$ corresponds to first applying the correspnding set-map $FY\to FZ$, and then the function to morphisms $Z\to Y$.
But this can also be presented as associating to each morphism $Y\to X$ the set of elements of $FY$ mapped to it, in which case the set-functions $FY\to FZ$ send elements associated to $Y\to X$ to elements associated to the composite $Z\to Y\to X$. This is exactly the data of a presheaf on $C|X$, i.e. an object of $(C|X)^\vee$.
Moreover, a natural transformation $F\Rightarrow G$ through which $F\Rightarrow X^\vee$ factors as $F\Rightarrow G\Rightarrow X^\vee$, consisting of morphisms $FY\to GY$, is then the same data that for each morphism $Y\to X$ sends the elements of $FY$ associated to it, to elements of $GY$ associated to it, all in a way compatible with pre-compositions with morphisms $Z\to Y$.
Under the above correspondence, the presheaf $X^\vee$, equipped with the identity natural transformation to itself, corresponds to $\mathbf 1|X$. Indeed, the former is the terminal object in $C^\vee|X^\vee$, the latter the terminal object in $(C|X)^\vee$. More explicitly, the identity natural transformation $X^\vee\Rightarrow X^\vee$ associates to each morphism $Y\to X$ the singleton containing it.
Moreover, subobjects of $X^\vee$ in $C^\vee$ are the same data as subobjects of the identity natural transformation $X^\vee\Rightarrow X^\vee$ considered as an object of $C^\vee|X^\vee$. Thus under the correspondence, they also are the same as subobjects of $\mathbf 1|X$.
Explcitly, a subobject of the presheaf $\mathbf 1|X$ that is an object of the category $(C|X)^\vee$ assigns to each morphism $Y\to X$ either the empty subset or the whole subset of the singleton assigned to $Y\to X$ by $\mathbf 1|X$ (these being the only subsets of the singleton), in such a way that for each morphism $Z\to Y$ there is an inclusion of the subset associated to $Y$ into the subset associated to $Z$.
Now for each $Y$, the morphisms $Y\to X$ that are assigned the whole subset of the singleton form a subset of all morphisms $Y\to X$. The condition that morphisms $Z\to Y$ induce an inclusion of subsets of the singletons is then equivalent to mapping the subset of morphisms $Y\to X$ to the subset of morphisms $Z\to X$ that are associated to whole subsets of the singleton.
But the latter is precisely a subobject (what the book calls a part) of the representable presheaf $X^\vee$ that is an object of the category $C^\vee$.
A subobject classifier, by definition, has morphisms into it corresponding to subojects of the domain of the morphism. Thus such a subobject corresponds to a natural transformation $X^\vee\Rightarrow\Omega$. The Yoneda lemma identifies natural transformations $X^\vee\Rightarrow\Omega$ with elements of the set $\Omega_X$ that is the value at the object $X$ of $\Omega$ as a presheaf on $C$.
We thus have correspondences between elements of the set $\Omega_X$, natural transformations $X^\vee\Rightarrow\Omega$, subobjects of $X^\vee$ in $C^\vee$, subobjects of $X^\vee\Rightarrow X^\vee$ in $C^\vee|X^\vee$, and subobjects of $\mathbf 1|X$ in $(C|X)^\vee$