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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 23rd 2015
• CommentRowNumber2.
• CommentAuthorNikolajK
• CommentTimeDec 23rd 2015

Makes me think of when is one thing equal to some other thing?.

The formulation of the second law on the Wikipedia page is something I haven’t encountered, $a\neq\neg a$.

Makes me wonder if $a\cong(a\to 0)$, e.g. in particular $*\cong(*\to 0)$, doesn’t make sense somewhere.

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeDec 24th 2015
• (edited Dec 25th 2015)

In the setting of a category with zero object you have $A\simeq (A\to 0)$. I guess one should point to AT-category.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeDec 24th 2015

David, could you explain what you meant in your last comment? What is meant by $A \to 0$?

A side remark is that any cartesian closed category with zero object $0$ is trivial (i.e., equivalent to the terminal category). For any object $A$ we have $A \times 0 \cong 0$ since $0$ is initial and $A \times -$ has a right adjoint; we also have $A \times 0 \cong A$ since $0$ is terminal. Thus $A \cong 0$ for every $A$.

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeDec 25th 2015
• (edited Dec 25th 2015)

Sorry, I was using Nikolaj’s notation (at least as I guessed what it meant): $A \to 0$ is the function type, ie $Hom(A,0)$…… ./… ….

….

:-(

Blerg, obviously wrong. Never mind, it was just some random emission… :-/ (Edit: Was travelling last night on the train at the end of a long day of travelling, and didn’t think before I posted)