  The $\gamma$-ray production mechanism and its localization in blazars are
still a matter of debate. The main goal of this paper is to constrain the
location of the high-energy emission in the blazar TXS 2013+370 and to study
the physical and geometrical properties of the inner jet region on sub-pc
scales. VLBI observations at 86 GHz and space-VLBI at 22 GHz allowed us to
image the jet base with an angular resolution of $\sim$0.4 pc. By employing
CLEAN imaging and Gaussian model-fitting, we performed a thorough kinematic
analysis, which provided estimates of the jet speed, orientation, and component
ejection times. Additionally, we studied the jet expansion profile and used the
information on the jet geometry to estimate the location of the jet apex. VLBI
data were combined with single-dish measurements to search for correlated
activity between the radio and $\gamma$-ray emission. The high-resolution VLBI
imaging revealed the existence of a spatially bent jet, described by moving and
stationary features. New jet features are observed to emerge from the core,
accompanied by flaring activity in radio bands and $\gamma$ rays. The analysis
of the transverse jet width profile constrains the location of the mm core to
lie $\leq$ 2 pc downstream of the jet apex, and also reveals the existence of a
transition from parabolic to conical jet expansion at a distance of $\sim$54 pc
from the core, corresponding to $\sim$1.5$\times$10$^{\rm 6}$ Schwarzschild
radii. The cross-correlation analysis reveals a strong correlation between the
radio and $\gamma$-ray data, with the 1 mm emission lagging $\sim$49 days
behind the $\gamma$ rays. Based on this, we infer that the high energy emission
is produced at a distance of $\sim$1 pc from the VLBI core, suggesting that the
seed photon fields for the external Compton mechanism originate either in the
dusty torus or in the broad-line region.
