Displacements

TL;DR

This paper introduces the concept of displacements in category theory, exploring their properties and implications for adjoint functors, model categories, and homotopy descent, with potential applications in algebraic geometry and homotopy theory.

Contribution

It defines displacements along morphisms, analyzes their existence in various contexts, and develops categorical lemmas for future applications in homotopy descent and geometric invariant theory.

Findings

Displacements generalize cocartesian liftings in category theory.

Existence of displacements relates to the existence of left adjoints.

Provides categorical lemmas for homotopy descent techniques.

Abstract

Given a functor $p:E \rightarrow B$ and an object $e \in E$ , we define a \emph{displacement} of $e$ along a morphism $\varepsilon: p(e) \rightarrow b$, as a map $e \rightarrow \nabla_\varepsilon(e)$ satisfying a universal property analogue to that of a \emph{cocartesian lifting} (pushforward) \emph{\`a la} B\'enabou-Grothendieck-Street. There are many difficulties in geometry that come from the fact that forgetful functors such as $p: Var(\mathbf{C}) \rightarrow Top $ don't have displacements of objects along arbitrary maps. And this can be already seen abstractly, since the existence of a left adjoint to $p$, can be reduced to the existence of all displacements of the initial object. However some \emph{schematization functors} exist as approximations. In a broader context, if $B$ is a model category and $p$ is a right adjoint, then the right-induced model category on $E$ exists if and only if all displacements along any trivial cofibration $\varepsilon$, are weak $p$-equivalences. In these notes we provide some categorical lemmas that will be necessary for future applications. The idea is to have a \emph{homotopy descent process} for \emph{elementary displacements} when $p$ has a \emph{presentation} as a $2$-pullback of a family $\{p_i: E_i \rightarrow B\}_{i\in J}$. When suitably applied it should lead to techniques similar to Mumford's GIT through homotopy theory (simplicial presheaves).