Virtual Triangles

Avishay Vaknin


Abstract

Motivated by Neeman's K-theory for triangulated categories, we are
studying virtual triangles in a triangulated category. A virtual
triangle is a triangle, which is a direct summand of an exact triangle,
where the other summand is a direct sum of trivial triangles. We define
exact triangle to be a triangle,  which any two of its maps belong to a
distinguished triangle, and a distinguished triangle on an identity map
to be a trivial triangle. A direct sum of trivial triangles is called a
splitting triangle, and a direct summand,
where the other summand splits,  is called an easy direct summand. So,
by definition, virtual triangles are closed under easy direct summands.
It turns out that they are also closed under direct sums, mapping cones,

and the t-structur truncation operation. This leads, using Neeman's
methods, to the virtual $K$-theory for triangulated categories. We have
in this theory, a version of the theorem of the heart, and it contains
Quillen's $K$-theory for the heart as a retract. Unlike Neeman's theory,

we don't know if this two $K$-theories coincide. The benefit of this
theory is, that it is functorial, and does not make any use of models,
in the sense of Waldhausen. We explain when pseudo triangles are all
virtual triangles,  where a pseudo triangle is a triangle, that goes to
a long exact sequence by any homology or cohomology functor. At the end,

we introduce the notion of fuzzy triangles, in order to construct a
virtual triangle, which is not exact. This shows, that exact triangles
are not closed under easy direct summands.