  Inclusive and differential cross-sections for the production of a top-quark
pair in association with a photon are measured with proton-proton collision
data corresponding to an integrated luminosity of 36.1 fb$^{-1}$, collected by
the ATLAS detector at the LHC in 2015 and 2016 at a centre-of-mass energy of 13
TeV. The measurements are performed in single-lepton and dilepton final states
in a fiducial volume. Events with exactly one photon, one or two leptons, a
channel-dependent minimum number of jets, and at least one $b$-jet are
selected. Neural network algorithms are used to separate the signal from the
backgrounds. The fiducial cross-sections are measured to be 521 $\pm$ 9(stat.)
$\pm$ 41(sys.) fb and 69 $\pm$ 3(stat.) $\pm$ 4(sys.) fb for the single-lepton
and dilepton channels, respectively. The differential cross-sections are
measured as a function of photon transverse momentum, photon absolute
pseudorapidity, and angular distance between the photon and its closest lepton
in both channels, as well as azimuthal opening angle and absolute
pseudorapidity difference between the two leptons in the dilepton channel. All
measurements are in agreement with the theoretical predictions.
