Returning to lattice theory, recall that the order relation between objects is represented by an arrow $a \rightarrow b$. Thus the lattice of two elements (0 and 1) is the same as the category 2.

Now in the topos Set, truth and falsity arise as follows. Truth is a given arrow from the one point set into {0,1}. Falseness is a characterising arrow in a basic pullback square, such that the square source is the initial object in the category, namely the empty set. In other words, both true and false are arrows into a special object $\Omega$. The action of turning true into false introduces the concept of complementation. Grothendieck’s great insight into topological spaces was to see that $\neg \neg U$ need not be $U$, and that this was a logical statement.

In replacing binary logic by ternary logic, it is natural to assume three arrows into an object $\Omega$. This suggests the replacement of pullback squares by cubes. Instead of a lattice $0 \rightarrow 1$ we begin with the next simplex, a triangle on the objects 0, 1 and 2. The triangle has a face, as do the square faces of a cube. Complementation must be replaced by a three-way swapping of basic truth values. This approach to 3-logos theory offers a new way of generalising topological spaces to higher categorical dimensions.

Observe that the faces (2-arrows) contribute to the weaker concept of complementation. In the trivalent corner of a spatial cube they play the role of dual time. It should not be surprising, therefore, to find that mass generation is linked to the sixth face of the tricategorical parity cube, the faces of which represent internalised edges of the Mac Lane pentagon.

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## Matti Pitkanen said,

May 12, 2007 @ 5:14 am

Dear Kea,

for some reason my comment did not go through. This has occurred already earlier a couple of times. So I try again.

If I have understood correctly, logical deductions could be seen as “category-theoretical reductionism” to initial objects by truth preserving paths or as reverse paths from terminal objects.

The following example concretizes Grothendienck’s vision. Consider the categories of sets and open sets in plane, call then Set and Open. In Set negation of statement is represented as the complement of set. In Open the interior of complement would represent it. Therefore plane with any number of punctures, which is open set, has always empty set as interior of complement so that negation of empty set is not unique and double negation can lead from one punctured plane to another punctured plane.

I encountered so called p-adic logics and matrix logics for some time ago and the problem of defining the negation or its generalization in a natural manner. For instance, should one replace negation with n-valued cyclic map? That would mean that initial and final objects are replaced by vertices of n-polygon so that one would have a new variant of category-theoretical reductionism!

Just thoughts;-)

## Kea said,

May 12, 2007 @ 9:53 pm

I encountered so called p-adic logics and matrix logics for some time ago and the problem of defining the negation or its generalization in a natural manner. For instance, should one replace negation with n-valued cyclic map?Hi Matti. It’s great to see this linking into your ideas. Yes, naturalness of the ‘higher topos’ definitions is the key here. From kneemo’s remarks, a cyclic generalisation of Heyting algebra would be a good place to start. By necessity, this means dealing with generalised complementation, and I agree it is important to keep Grothendieck’s vision in mind while doing this. Heh, I wish I could discipline myself to struggle through his whole letter to Quillen…

Re BLOGGER: they are being painful, but they keep promising to sort this out.

## L. Riofrio said,

May 14, 2007 @ 8:03 pm

This could lead to sdome princviple behind the time we experience, even “time’s arrows.”