objects are closed, oriented $3$-dimensional bifolds $\check Y$ with bifold metrics, and in which the morphisms are isomorphism classes of oriented $4$-dimensional bifolds $\check X$ with boundary. Equivalently, we have a category in which the objects are $3$-manifolds $(Y,K)$ with embedded webs, and the morphisms are $4$-manifolds $(X,\Phi)$ with embedded foams.

#### Bifold connections.

By a *bifold connection* over a bifold $\check X$, we mean an $\SO(3)$ orbifold vector bundle $E\to\check X$ equipped with an orbifold $\SO(3)$ connection $A$, subject to the constraint that at each point $x$ where $H_{x}$ has order $2$, the local action of $H_{x}$ on the $\SO(3)$ fiber is non-trivial. This condition determines the local model uniquely at other orbifold points. In particular, if $\check X$ is $4$-dimensional and $x$ belongs to a seam of the corresponding foam, so that $H_{x}$ is the Klein $4$-group, then the representation of $H_{x}$ on the $\SO(3)$ fiber is the inclusion of the standard Klein $4$-group $V\subset \SO(3)$.

#### Marking data.

Bifold connections may have non-trivial automorphisms. For example, if the monodromy group of the connection is the $4$-group $V$, then the automorphism group is also $V$. In order to have objects without automorphisms, we introduce *marked* bifold connections.

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