# On factorizations of graphical maps

## TL;DR

This paper explores the categorical structures underlying infinity properads, establishing the graphical category as an Eilenberg-Zilber category and developing model structures for Segal properads and wheeled properads.

## Contribution

It identifies the graphical category as an Eilenberg-Zilber category and introduces model structures for Segal properads and wheeled properads, advancing the categorical understanding of these algebraic structures.

## Key findings

- Graphical category is an Eilenberg-Zilber category.
- Wheeled graphical category can be made into a generalized Reedy category.
- Model structures for Segal properads and wheeled properads are constructed.

## Abstract

We study the categories governing infinity (wheeled) properads. The graphical category, which was already known to be generalized Reedy, is in fact an Eilenberg-Zilber category. A minor alteration to the definition of the wheeled graphical category allows us to show that it is a generalized Reedy category. Finally, we present model structures for Segal properads and Segal wheeled properads.