# What is the spectral category?

## TL;DR

This paper constructs a spectral category from a given category with specific monomorphism properties, demonstrating it has finite limits and preserves them, especially in the context of normal categories.

## Contribution

It introduces a method to build spectral categories using pullback stable essential monomorphisms, expanding the understanding of their structure and properties.

## Key findings

- Spectral category $ ext{Spec}( ext{C}, ext{S})$ has finite limits.
- Canonical functor preserves finite limits.
- In normal categories, essential monomorphisms coincide with subobject-essential monomorphisms.

## Abstract

For a category $\mathcal{C}$ with finite limits and a class $\mathcal{S}$ of monomorphisms in $\mathcal{C}$ that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback stable $\mathcal{S}$-essential monomorphisms in $\mathcal{C}$ to construct a spectral category $\mathrm{Spec}(\mathcal{C},\mathcal{S})$. We show that it has finite limits and that the canonical functor $\mathcal{C}\to \mathrm{Spec}(\mathcal{C},\mathcal{S})$ preserves finite limits. When $\mathcal{C}$ is a normal category, assuming for simplicity that $\mathcal{S}$ is the class of all monomorphisms in $\mathcal{C}$, we show that pullback stable $\mathcal{S}$-essential monomorphisms are the same as what we call subobject-essential monomorphisms.