# Higher Homotopic Distance

## TL;DR

This paper explores the properties of higher homotopic distance, its relation to topological invariants like $ ext{cat}$, $ ext{secat}$, and higher topological complexity, providing new proofs and insights.

## Contribution

It introduces important properties of higher homotopic distance and establishes conditions linking it to classical topological invariants, offering alternative proofs for related theorems.

## Key findings

- Higher homotopic distance properties are characterized.
- Conditions for equality between homotopic distance and invariants like $ ext{cat}$, $ ext{secat}$, and topological complexity are identified.
- Alternative proofs for $ ext{TC}_n$-related theorems using higher homotopic distance are provided.

## Abstract

The concept of homotopic distance and its higher analog are introduced in [6]. In this paper we introduce some important properties of higher homotopic distance, investigate the conditions under which $\cat$, $\secat$ and higher dimensional topological complexity are equal to the higher homotopic distance, and give alternative proofs, using higher homotopic distance, to some $\TC_n$-related theorems.