Yes, I think Girard introduced the lollipop along with some other strange symbols like the upside-down and.

In logic and physics we mainly use _braided_ and _symmetric_ monoidal closed categories, and I wrote an introduction to those here:

* John Baez and Mike Stay, [Physics, topology, logic and computation: a Rosetta Stone](http://arxiv.org/abs/0903.0340) in _New Structures for Physics_, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95-172.

The first time I saw really interesting applications of closed monoidal categories that weren't symmetric or even braided was in Lambek's work on linguistics. When you don't have a braiding isomorphism \$$x \otimes y \cong y \otimes x\$$ you need to distinguish between left and right closed monoidal categories, which have two operations x⊸y and y⟜x, or in other words x\y and y/x. Lambek uses these to describe grammatical operations! His **pregroups** are monoidal preorders that are both left and right closed. My student Jade and another student at ACT2018 wrote a pretty good introduction to this idea, which is pretty simple:

* Jade Master and Cory Griffith, [Linguistics using category theory](https://golem.ph.utexas.edu/category/2018/02/linguistics_using_category_the.html), _The n-Category Café_, 6 February 2018.