# On the Faithfulness of 1-dimensional Topological Quantum Field Theories

## TL;DR

This paper investigates the faithfulness of 1-dimensional topological quantum field theories, demonstrating that both strict and strong variants are faithful functors, thereby clarifying their structural properties.

## Contribution

It provides a rigorous proof that both strict and strong 1-dimensional TQFTs are faithful, enhancing understanding of their categorical and algebraic structures.

## Key findings

- Both strict and strong 1D TQFTs are faithful functors.
- Strict 1D TQFTs are symmetric monoidal functors to matrices.
- Strong 1D TQFTs are symmetric monoidal functors to finite-dimensional vector spaces.

## Abstract

This paper explores 1-dimensional topological quantum field theories. We separately deal with strict and strong 1-dimensional topological quantum field theories. The strict one is regarded as a symmetric monoidal functor between the category of 1-cobordisms and the category of matrices, and the strong one is a symmetric monoidal functor between the category of 1-cobordisms and the category of finite dimensional vector spaces. It has been proved that both strict and strong 1-dimensional topological quantum field theories are faithful.