  Broad-band images of the solar chromosphere in the Ca II H&K line cores
around active regions are covered with fine bright elongated structures called
bright fibrils. The mechanisms that form these structures and cause them to
appear bright are still unknown. We aim to investigate the physical properties,
such as temperature, line-of-sight velocity, and microturbulence, in the
atmosphere that produces bright fibrils and to compare those to the properties
of their surrounding atmosphere. We used simultaneous observations of a plage
region in Fe I 6301-2 \r{A}, Ca II 8542 \r{A}, Ca II K, and H$\alpha$ acquired
by the CRISP and CHROMIS instruments on the Swedish 1-m Solar Telescope. We
manually selected a sample of 282 Ca II K bright fibrils. We compared the
appearance of the fibrils in our sample to the Ca II 8542 \r{A} and H$\alpha$
data. We performed non-local thermodynamic equilibrium (non-LTE) inversions
using the inversion code STiC on the Fe I 6301-2 \r{A}, Ca II 8542 \r{A}, Ca II
K lines to infer the physical properties of the atmosphere. The line profiles
in bright fibrils have a higher intensity in their K$_2$ peaks compared to
profiles formed in the surrounding atmosphere. The inversion results show that
the atmosphere in fibrils is on average $100-200$~K hotter at an optical depth
log$(\tau) = -4.3$ compared to their surroundings. The line-of-sight velocity
at chromospheric heights in the fibrils does not show any preference towards
upflows or downflows. The microturbulence in the fibrils is on average 0.5 km
s$^{-1}$ higher compared to their surroundings. Our results suggest that the
fibrils have a limited extent in height, and they should be viewed as hot
threads pervading the chromosphere.
