# Endomorphism operads of functors

## TL;DR

This paper investigates the endomorphism operad of functors, especially the forgetful functor from algebras over an operad, revealing when it recovers the original operad across different categories.

## Contribution

It demonstrates that the endomorphism operad of the forgetful functor recovers the original operad in vector spaces over infinite fields, but not in finite fields or sets, with several examples computed.

## Key findings

- Recovers operad in vector spaces over infinite fields
- Fails to recover operad in finite fields and sets
- Provides multiple computed examples

## Abstract

We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been implicitly studied. We ask whether the endomorphism operad of the forgetful functor from algebras over an operad to the ground category recovers that operad. The answer is positive for operads in vector spaces over an infinite field, but negative both in vector spaces over finite fields and in sets. Several examples are computed.