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Free rigid commutative algebras

Maxime Ramzi

Abstract

We describe free rigid commutative algebras in $2$-presentably symmetric monoidal $(\infty,2)$-categories as oplax colimits over the $1$-dimensional framed cobordism category. The special case of the $(\infty,2)$-category $\mathrm{Pr}^\mathrm{L}$ itself provides a description of the free symmetric monoidal $(\infty,1)$-category with duals on a given $(\infty,1)$-category, while the case of $\mathrm{Mod}_{\mathcal{V}}(\mathrm{Pr}^\mathrm{L})$ provides a description of a similar object in the $\mathcal{V}$-enriched context, for $\mathcal{V}$ a presentably symmetric monoidal $(\infty,1)$-category. As a byproduct, we obtain new proofs of some results about rigidification of locally rigid categories, as well as a proof that any rigid category over $\mathrm{Sp}$ embeds into a compactly-rigidly generated one.

Free rigid commutative algebras

Abstract

We describe free rigid commutative algebras in -presentably symmetric monoidal -categories as oplax colimits over the -dimensional framed cobordism category. The special case of the -category itself provides a description of the free symmetric monoidal -category with duals on a given -category, while the case of provides a description of a similar object in the -enriched context, for a presentably symmetric monoidal -category. As a byproduct, we obtain new proofs of some results about rigidification of locally rigid categories, as well as a proof that any rigid category over embeds into a compactly-rigidly generated one.

Paper Structure

This paper contains 9 sections, 46 theorems, 50 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Left adjoints
  3. Rigid algebras
  4. Oplax colimits
  5. The construction
  6. The proof
  7. Some calculations
  8. Consequences for locally rigid algebras
  9. Oplax colimits of lax symmetric monoidal functors

Key Result

Theorem 1

Let $\mathbf{B}$ be a weakly $2$-presentably symmetric monoidal $(\infty,2)$-category, and $b\in \mathbf{B}$ be a dualizable object. There exists a free rigid commutative algebra on $b$ in $\mathbf{B}$, and it is given by a canonical commutative algebra structure on the oplax colimit of the unique symmetric monoidal functor $\mathrm{Cob}^{\mathrm{1d,or}}\to \mathbf{B}$ sending the positively orie

Figures (1)

  • Figure 1: Farbstudie - Quadrate und konzentrische Ringe, W. Kandinsky

Theorems & Definitions (118)

  • Example
  • Theorem : \ref{['thm:maintext']}
  • Example
  • Example
  • Corollary : \ref{['cor:End1']}
  • Example
  • Corollary : \ref{['cor:ffrigidemb']}
  • Remark 1
  • Remark
  • Example 1
  • ...and 108 more