| Summary: | The higher category theory can be employed to generalize the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula> action to the so-called <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mi>B</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula> action, by passing from the notion of a gauge group to the notion of a gauge 3-group. The theory of scalar electrodynamics coupled to Einstein–Cartan gravity can be formulated as a constrained <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mi>B</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula> theory for a specific choice of the gauge 3-group. The complete Hamiltonian analysis of the <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mi>B</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula> action for the choice of a Lie 3-group corresponding to scalar electrodynamics is performed. This analysis is the first step towards a canonical quantization of a <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mi>B</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula> theory, an important stepping stone for the quantization of the complete scalar electrodynamics coupled to Einstein–Cartan gravity formulated as a <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mi>B</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula> action with suitable simplicity constraints. It is shown that the resulting dynamic constraints eliminate all propagating degrees of freedom, i.e., the <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mi>B</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula> theory for this choice of a 3-group is a topological field theory, as expected.
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