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The subobject decomposition in enveloping tensor categories

Friedrich Knop

Abstract

To every regular category $\mathcal{A}$ equipped with a degree function $δ$ one can attach a pseudo-abelian tensor category $\mathcal{T}(\mathcal{A},δ)$. We show that the generating objects of $\mathcal{T}$ decompose canonically as a direct sum. In this paper we calculate morphisms, compositions of morphisms and tensor products of the summands. As a special case we recover the original construction of Deligne's category $\operatorname{Rep} S_t$.

The subobject decomposition in enveloping tensor categories

Abstract

To every regular category equipped with a degree function one can attach a pseudo-abelian tensor category . We show that the generating objects of decompose canonically as a direct sum. In this paper we calculate morphisms, compositions of morphisms and tensor products of the summands. As a special case we recover the original construction of Deligne's category .

Paper Structure

This paper contains 6 sections, 15 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. The category ${\mathcal{T}}({\mathcal{A}},\delta)$
  3. An alternate construction of ${\mathcal{T}}({\mathcal{A}},\delta)$
  4. The subobject decomposition
  5. The subobject construction of ${\mathcal{T}}({\mathcal{A}},\delta)$
  6. A formula for exact Mal'tsev categories

Key Result

Theorem 1.1

Let ${\mathcal{A}}$ be a subobject finite, regular category, let $\delta$ be a degree function on ${\mathcal{A}}$ and ${\mathcal{T}}:={\mathcal{T}}({\mathcal{A}},\delta)$. For all objects $x$ and $y$ of ${\mathcal{A}}$ let $R(x,y)$ be the set of subobjects of $x\times y$ such that both projections $ for all $x$. Moreover, the objects $[x]^*$ have the following properties: In particular, ${\mathca

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Definition 1
  • Remark 3
  • Definition 2
  • Remark 4
  • Definition 3
  • Definition 4
  • Theorem 3.1
  • ...and 30 more