# Representability theorems, up to homotopy

## TL;DR

This paper proves two homotopy-based representability theorems for presheaves valued in a closed symmetric combinatorial model category, with applications to recognizing mapping spaces.

## Contribution

It introduces two new representability theorems up to homotopy, extending classical results to a homotopical setting.

## Key findings

- Proves a homotopy version of Freyd's representability theorem.
- Establishes a homotopy version of Brown's representability theorem.
- Provides a recognition principle for mapping spaces.

## Abstract

We prove two representability theorems, up to homotopy, for presheaves taking values in a closed symmetric combinatorial model category \cat V. The first theorem resembles the Freyd representability theorem, the second theorem is closer to the Brown representability theorem. As an application we discuss a recognition principle for mapping spaces.