# The homotopy theory of operad subcategories

## TL;DR

This paper investigates the homotopy theory of subcategories of topological operads with a focus on unitary and truncated operads, establishing model structures and equivalences of derived mapping spaces.

## Contribution

It proves that the subcategory of unitary operads inherits a model structure and that the derived mapping spaces are homotopy equivalent to those in all operads, extending to truncated operads.

## Key findings

- The subcategory of unitary operads admits a model structure.
- Derived mapping spaces in unitary operads are homotopy equivalent to those in all operads.
- Results extend to k-truncated operads for any fixed arity bound k.

## Abstract

We study the subcategory of topological operads $P$ such that $P(0) = *$ (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping spaces associated to our model category of unitary operads are homotopy equivalent to the standard derived operad mapping spaces, which we form in the model category of all operads in topological spaces. We prove that analogous statements hold for the subcategory of $k$-truncated unitary operads within the model category of all $k$-truncated operads, for any fixed arity bound $k\geq 1$, where a $k$-truncated operad denotes an operad that is defined up to arity $k$.