We study the lattice of all Borel clones on $2 = \{0,1\}$: classes of Borel functions $f : 2^n \to 2$, $n \le ω$, which are closed under composition and include all projections. This is a natural extension to countable arities of Post's 1941 classification of all clones of finitary Boolean functions. Every Borel clone restricts to a finitary clone, yielding a "projection" from the lattice of all Borel clones to Post's lattice. It is well-known that each finitary clone of affine mod 2 functions admits a unique extension to a Borel clone. We show that over each finitary clone containing either both $\wedge, \vee$, or the 2-out-of-3 median operation, there lie at least 2 but only finitely many Borel clones. Over the remaining clones in Post's lattice, we give only a partial classification of the Borel extensions, and present some evidence that the full structure may be quite complicated.

Low-light image enhancement (LLIE) can be reformulated as an image-specific curve estimation (CE) problem. Traditional CE-based methods struggle with issues such as uniform processing across different regions, static parameter estimation, and lack of effective global semantic enhancement. To address these limitations, we propose a novel unsupervised learning framework, Patch-wise Dynamic Curve Estimation (PDCE), which dynamically adjusts and optimizes enhancement curves according to local patch brightness and the iteration process. Specifically, we present a Vision-Language Curve Discriminator (VLCD), which dynamically determines the curve type for each patch, avoiding uniformly applying the curve on the whole image. We introduce a Curve Parameter Estimator (CPE), which dynamically updates curve parameters and adjusts enhancement effects based on the output of the previous iteration. Furthermore, we design a Visual State Space-based Semantic Enhancement Module (VSEM), which captures global receptive fields and enriches semantic features through the Mamba-based U-Net architecture. Extensive experimental results show the superiority of our PDCE over state-of-the-art methods for LLIE.

Low-light images are commonly present due to imaging factors such as insufficient light, night shooting and back lit. Existing low-light image enhancement (LLIE) methods typically rely on a low-light input image for enhancement, which seldom leverage information contained in its high-light counterpart to restore image structures and handle complex lighting, leading to unsatisfactory image quality. In view of this, in this paper we propose a Collaborative Dual-Branch Spatial-Frequency Enhancement Network (CDSE-Net). Specifically, we apply the inversion operation to low-light images to self-generate high-light images and build a collaborative dual-branch network which enhances images sequentially in spatial and frequency domains. In the spatial domain, we leverage adaptive curve estimation and multi-direction convolutions to restore lightness and structure information, respectively. In the frequency domain, we perform amplitude interactions on dual-branch images and 1×1 convolution on phase features to adjust image lightness and structures, respectively. Finally, we introduce an illumination-aware attention module to fuse two branches. Experiments on several widely used datasets quantitatively and qualitatively demonstrate the advantages of our network over state-of-the-art methods for LLIE.

Abstract We prove that the category $\mathsf {SBor}$ of standard Borel spaces is the (bi-)initial object in the 2-category of countably complete Boolean (countably) extensive categories. This means that $\mathsf {SBor}$ is the universal category admitting some familiar algebraic operations of countable arity (e.g., countable products and unions) obeying some simple compatibility conditions (e.g., products distribute over disjoint unions). More generally, for any infinite regular cardinal $\kappa $ , the dual of the category $\kappa \mathsf {Bool}_{\kappa }$ of $\kappa $ -presented $\kappa $ -complete Boolean algebras is (bi-)initial in the 2-category of $\kappa $ -complete Boolean ( $\kappa $ -)extensive categories.

Abstract A category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram has a cocone. We show that for a finitely generated category I, the following are equivalent: (i) every I-shaped diagram in a category with the AP and the JEP has a cocone; (ii) every I-shaped diagram in the category of sets and injections has a cocone; (iii) a certain canonically defined category ${\cal L}\left( {\bf{I}} \right)$ of “paths” in I has only idempotent endomorphisms. When I is a finite poset, these are further equivalent to: (iv) every upward-closed subset of I is simply-connected; (v) I can be built inductively via some simple rules. Our proof also shows that these conditions are decidable for finite I.