# Cofibrations in the Category of Frolicher Spaces. Part I

## TL;DR

This paper introduces a new concept of cofibrations in the category of Fr"olicher spaces, facilitating smooth homotopy extensions and relating them to neighborhood deformation retracts, with applications to constructing the right Puppe sequence.

## Contribution

It defines smooth cofibrations in Fr"olicher spaces using flattened intervals and relates them to neighborhood deformation retracts, advancing the theory of smooth homotopy.

## Key findings

- Smooth cofibrations enable easier smooth homotopy extensions.
- A characterization of neighborhood deformation retracts in terms of cofibrations.
- Construction of the right Puppe sequence in this context.

## Abstract

Cofibrations are defined in the category of Fr\"olicher spaces by weakening the analog of the classical definition to enable smooth homotopy extensions to be more easily constructed, using flattened unit intervals. We later relate smooth cofibrations to smooth neighborhood deformation retracts. The notion of smooth neighborhood deformation retract gives rise to an analogous result that a closed Fr\"olicher subspace $A$ of the Fr\"olicher space $X$ is a smooth neighborhood deformation retract of $X$ if and only if the inclusion $i: A\hookrightarrow X$ comes from a certain subclass of cofibrations. As an application we construct the right Puppe sequence.