Schemesome

Schemesome is a term that appears in informal mathematical and theoretical discussions as a placeholder for a generalized geometric object that extends the notion of a scheme from algebraic geometry. Unlike standard schemes, a schemesome is described with varying definitions across sources and lacks formal consensus.

Origins of the term are informal, often arising in online forums, workshop notes, or speculative papers where researchers seek a unifying framework that could encompass schemes, stacks, and derived or spectral geometries. Because there is no widely accepted formalization, the concept is typically introduced with hedges and context-dependent details.

Definitions and variants are diverse. Some readers imagine a schemesome as a site-based object carrying local algebraic data with gluing rules more flexible than those of ordinary schemes, potentially incorporating higher-categorical, derived, or noncommutative data. Others propose a topos-theoretic or Higher-Categorical formulation that subsumes schemes and related geometric objects as special cases. A common thread is the idea of local patches that resemble schemes but allow broader glueing and structure, without committing to a single precise axiomatization.

Examples and use cases are largely speculative. In discussions, schemesome objects are used as models for generalized moduli spaces, noncommutative spaces, or derived enhancements, serving as a testbed for unifying approaches in geometry and mathematical physics. However, due to the lack of standard definitions, concrete examples are informal and vary between authors.

Relation to established concepts: schemes are often viewed as the base case or a special instance within schemesome frameworks, with schemesome intended to include schemes, algebraic stacks, and certain derived geometries under a common umbrella. Status: the term remains exploratory and not part of formal, widely adopted theory.

See also: scheme, algebraic geometry, sheaf theory, topos theory, stacks, derived geometry.