A note on the composition product of symmetric sequences

The more legs the merrier: A new composition for symmetric (multi-)lenses

Lenses are a category theoretic construct and are used in a wide variety of applications. Symmetric lenses compose to, of course, form new symmetric lenses. Symmetric lenses are usually represented as spans of asymmetric lenses. In many applications, the fact that a symmetric lens might also be represented as a cospan of asymmetric lenses is important, especially for implementation purposes. However, the composition of symmetric lenses does not preserve the property that the lenses can be represented by cospans -- two such symmetric lenses may (and frequently do) compose to form a symmetric le…

一个（不那么）简短的注记：对称幺半范畴的各种定义的等价性

A (not so) short note: the equivalence of various notions of symmetric monoidal category

本文证明了经典偏向对称幺半范畴、无偏对称幺半范畴和同伦对称幺半范畴在精确意义下等价。

A note on the Wehrheim-Woodward category

Wehrheim and Woodward have shown how to embed all the canonical relations between symplectic manifolds into a category in which the composition is the usual one when transversality and embedding assumptions are satisfied. A morphism in their category is an equivalence class of composable sequences of canonical relations, with composition given by concatenation. In this note, we show that every such morphism is represented by a sequence consisting of just two relations, one of them a reduction and the other a coreduction.

Traces in symmetric monoidal categories

The purpose of this expository note is to describe duality and trace in a symmetric monoidal category, along with important properties (including naturality and functoriality), and to give as many examples as possible. Among other things, this note is intended as background for the generalizations to the context of bicategories and indexed monoidal categories.