arXiv subjects

Samuel Leblanc

Publications and source records attributed to Samuel Leblanc.

4 recordsLinked to original sources

arXiv2026

Initial Subcategories via Free Presentations of the Constant Functor

Let $\mathcal{C}$ be a small category. We characterize the full initial subcategories of $\mathcal{C}$ as precisely those whose representables give a free presentation of the constant functor $Δ_{\mathcal{C}}\mathbb{K}:\mathcal{C}\to\mathrm{Vect}$. Such a presentation yields a formula for the limit of any functor $\mathcal{C} \to \mathrm{Vect}$ as the kernel of a morphism between products of evaluations. When the linearized endomorphism ring of every object is local, we show that a minimal free presentation, whenever one exists, determines the unique minimal full initial subcategory, up to equivalence. Bass's characterization of perfect rings gives a sufficient condition for the existence of such presentations. In particular, every finite category with no nonidentity endomorphisms, such as a finite poset, contains a unique minimal initial full subcategory. In the poset case, our results recover the minimal initial subposet constructed in [BDL27] and [DL26]. When $\mathcal{C}$ is the free category on a finite quiver, we give an explicit construction of its minimal initial full subcategories, show that every such subcategory arises from this construction, and provide algorithms with time-complexity bounds.

math.CT↗

arXiv2026

Generalized Rank via Minimal Subposet

Let $\mathcal{C}$ be a small, connected category with finite hom-sets. We show that if the embedding of a connected subcategory $\mathcal{J}$ is both initial and final, then the restriction of any $\mathcal{C}$-module along $\mathcal{J}$ preserves the generalized rank-or equivalently, the multiplicity of the ``entire" interval modules for $\mathcal{C}$ and $\mathcal{J}$. Conversely, we prove that this property characterizes initial and final embeddings when both $\mathcal{C}$ and $\mathcal{J}$ are posets satisfying certain mild constraints and the embedding is full. For $\mathcal{C}$ a poset under these conditions, we describe the minimal full subposet whose embedding is initial or final. This generalizes an observation made by Dey and Lesnick. We also extend a result of Kinser on the generalized rank invariant to small categories.

math.RT↗

arXiv2025

Batalin-Vilkovisky structure on Hochschild cohomology with coefficients in the dual algebra

We prove that Hochschild cohomology with coefficients in $A^*=\Hom_k(A,k)$ under conditions on the algebra structure of $A^*$ is a Batalin-Vilkovisky algebra. We also show that for symmetric and Frobenius algebras, this recovers the known BV-structures in Hochschild cohomology with coefficients in $A$ but admits an easy-to-describe BV-operator. Finally, we show that for monomial algebras $A = kQ/\langle T \rangle$, the Hochschild cohomology with coefficients in $A^*$ is always a Batalin-Vilkovisky algebra.

math.KT↗

arXiv2024

Hidden Activations Are Not Enough: A General Approach to Neural Network Predictions

We introduce a novel mathematical framework for analyzing neural networks using tools from quiver representation theory. This framework enables us to quantify the similarity between a new data sample and the training data, as perceived by the neural network. By leveraging the induced quiver representation of a data sample, we capture more information than traditional hidden layer outputs. This quiver representation abstracts away the complexity of the computations of the forward pass into a single matrix, allowing us to employ simple geometric and statistical arguments in a matrix space to study neural network predictions. Our mathematical results are architecture-agnostic and task-agnostic, making them broadly applicable. As proof of concept experiments, we apply our results for the MNIST and FashionMNIST datasets on the problem of detecting adversarial examples on different MLP architectures and several adversarial attack methods. Our experiments can be reproduced with our \href{https://github.com/MarcoArmenta/Hidden-Activations-are-not-Enough}{publicly available repository}.

cs.LG↗