1. Terminologically, the taco is a determinate monoid that abstracts the multiplicative structure inherent in non-degenerate series of essential localisations that resolve/sublate each other. Hence, strictly speaking the question should be whether the monads you mention can occur as monads induced by essential localizations in such a series. Concerning the internal hom, I think the answer is yes, because open localisations are of this form for subterminal objects $X$ and these are essential and I know of no reason why they shouldn't be resolved/sublated in general (the identity monad resolves trivially any essential localisation). Concerning "product with X" I don't know (of course for X=1 this is trivially essential and as identity even sublating) but at Science of logic you find quite a few adjoint cylinders possibly with your favorite monad among it.

2. The double negation monad i.e. the Booleanization of a topos (provided it is essential) typically sublates the trivial "empty" subtopos. This is detailed at Aufhebung where also some sufficient conditions are given for this to occur. There is also given a reference to a paper of Zaks, Riehl et al. that work out the complete Aufhebungs relation for three toposes of combinatorial topology illustrating that there are toposes where all contradictions = adjoint cylinders are sublated whence that a four series of resolving essential localizations need not contain the double negation subtopos since the chains of resolution/sublation become arbitrarily long in these toposes.

3. In Hegel speak there is only the term of Aufhebung i.e. what is abstracted here is the passage from contradictory category to a new sublating contradictory category. The localisations are four presumably because besides the pair where one sublates the other there is also the ambient topos and a third subtopos that also resolves the first contradiction but contains the sublating subtopos (i.e. this approximates the "minimality condition" in Aufhebung: dialectics replaces a contradictory concept with a new concept in which the original contradiction disappears by making the smallest possible conceptual change; all later occurring concepts also have the original opposition resolved but stand in a less direct relation to it since they might resolve other contradictions ). ]]>

1. Suppose we are in a cartesian closed category. Does the monad [X, -] (internal hom) or the monad "product with X" ever get incorporated into a taco?

2. Is one of the essential localizations the essential localization at the double negation monad?

3. What does each of the four essential localizations correspond to in Hegel speak? ]]>

Whence in my opinion what this entry needs more urgently is to give the multiplication table a reality check (with automatic proof checker) in order to verify that it is indeed correct, then give the taco monoid a neater representation in terms of generators and relations (which I had tried unsuccessfully at the time of writing the entry though not too hard after having spent already some time on tediously calculating the multiplication table by hand). The next step then would be to stuff out the monoid taco with the additional 2-categorical data present in adjoint situations generally to a monoidal category and then write out what this monoidal category TACO yields for a Licata-Shulman-Reed adjoint logic when used as the 2-category of modes. Obviously, an energetic person is needed here! ]]>

Could someone add examples on this page? It is really quite exciting, but I am trying to understand some examples, from either ontology or mathematics.

Here are some monad comonad pairs (adjoint cylinders):

provable that x, not provable that not x

knowable that x, not knowable that not x (functoriality is a bit like modus ponens, counit is facticity, the multiplication says that knowable x -> knowable knowable x, which is perhaps disputable)

possible, necessary

sometimes, always

exists, forall

Do these adjoint cylinders produce tacos on their corresponding categories?

]]>I would be most grateful if anyone can share the 1989 taco paper… I can be reached here (scr.im email address link). Thanks!

]]>@David C : I agree, though perhaps just the actual diagram, with the text quoted nearby. I should thank John B for supplying me with the article; I was hesitant to do so in a more public venue.

EDIT: I just went and pasted in the picture as I put in on G+. If someone else wants to pretty it up then go ahead.

]]>Hegel has some strange birthday gifts this year: math encyclopedia entry on a taco, rap album…

]]>Unfortunately, type-setting the taco took so long that I haven’t found the time until now to give public warning here that the multiplication table there is still longing very much for an automated proof checker (or an attentive reader).

The underlying configuration of adjoint modalities might also find smoother expression with the help of some of the n-adjoint modal logicians here.

]]>We should include that picture that you, David R., showed on g+, explaining the name ’taco’.

]]>I do wonder if we should cite *Burritos for the hungry mathematician,* by Ed Morehouse p46 of http://sigbovik.org/2015/proceedings.pdf (as ’in popular culture’? ;-P)

Thomas Holder is busy creating Hegelian taco as I write.

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