The purpose of this article is to compare several versions of bivariant algebraic
cobordism constructed previously by the author and others. In particular, we show
that a simple construction based on the universal precobordism theory of Annala
and Yokura agrees with the more complicated theory of bivariant derived
algebraic cobordism constructed earlier by the author, and that both of these
theories admit a Grothendieck transformation to operational cobordism
constructed by Luis González and Karu over fields of characteristic 0. The
proofs are partly based on convenient universal characterizations of several
cobordism theories, which should be of independent interest. Using similar
techniques, we also strengthen a result of Vezzosi on operational derived
-theory.
In the appendix, we give a detailed construction of virtual pullbacks in algebraic
bordism, filling the gaps in the construction of Lowrey and Schürg.
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