# nLab trace-class map

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

## Models

#### Higher category theory

higher category theory

# Contents

## Definition

Given a closed monoidal $(\infty, 1)$-category $\mathcal{C}$, with unit $I \in \mathcal{C}$, tensor product $(-)\otimes (-):\mathcal{C} \times \mathcal{C} \to \mathcal{C}$, and internal hom $[(-), (-)]:\mathcal{C} \times \mathcal{C} \to \mathcal{C}$, let $a \in \mathcal{C}$ and $b \in \mathcal{C}$ be objects in $\mathcal{C}$. Let $f:a \to b$ be a map. $f$ is a trace-class map or nuclear map if $f$ lies in the image of the natural map $\mathrm{Hom}(I, [a, I] \otimes b) \to \mathrm{Hom}(a, b)$.