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We present the Bidimensional Exploration of the warm-Temperature Ionised gaS (BETIS) project, designed for the spatial and spectral study of the diffuse ionised gas (DIG) in a selection of nearby spiral galaxies observed with the MUSE integral-field spectrograph. Our primary objective is to investigate the various ionisation mechanisms at play within the DIG. We analysed the distribution of high- and low-ionisation species in the optical spectra of the sample on a spatially resolved basis. We introduced a new methodology for spectroscopically defining the DIG, optimised for galaxies of different resolutions. We also examined the suitability of using Ha equivalent width (EWha) as a proxy for defining the DIG and its associated ionisation regime. The average radial distribution of the main line ratios are enhanced in the DIG. It follows similar trends between the DIG regime and the HII regions, as well as the Ha surface brightness, indicating a correlation between the ionisation of these species in both the DIG and the HII regions. The DIG loci in diagnostic diagrams are found within the line ratios that correspond to photoionisation due to the star formation. There is a noticeable offset correspondent to fast shocks. However, an individual diagnosis performed for each galaxy reveals that all the DIG in these galaxies can be attributed to photoionisation from star formation. The offset is primarily due to the contribution of Seyfert galaxies in our sample, which is closely aligned with models of ionisation from fast shocks and galactic outflows, thus mimicking the DIG emission. Our results indicate that galaxies exhibiting active galactic nucleus activity should be considered separately when conducting a general analysis of the DIG ionisation mechanisms, since this emission is indistinguishable from high-excitation DIG., Comment: 21 pages, 15 figures, A&A accepted

This is the second paper in a series using data from tens of thousands S0 galaxies of the local Universe ($z\lesssim 0.1$) retrieved from the NASA-Sloan Atlas. It builds on the outcomes of the previous work, which introduced a new classification scheme for these objects based on the principal component analysis (PCA) of their optical spectrum and its projections on to the first two eigenvectors or principal components (the PC1$\unicode{x2013}$PC2 diagram). We provide a comprehensive characterization of the activity of present-day S0s throughout both the broad-band PC1$\unicode{x2013}$PC2 spectral classifier and the conventional narrow-line BPT/WHAN ones, contrasting the different types of activity classes they define, and present an alternative diagram that exploits the concordance between WHAN and PCA demarcations. The analysis is extended to the mid-infrared, radio and X-ray wavelengths by crossmatching our core sample with data from the WISE, FIRST, XMM$\unicode{x2013}$Newton, and Chandra surveys. This has allowed us to carry out a thorough comparison of the most important activity diagnostics in the literature over different wavebands, discuss their similarities and differences, and explore the connections between them and with parameters related to star formation and black hole accretion. In particular, we find evidence that the bulk of nebular emission from radio and X-ray detected S0$\unicode{x2013}$Seyfert and LINER systems is not driven by star birth, while the dominant ionising radiation for a number of LINERs might come from post-AGB stars. These and other outcomes from the present work should be transferable to other morphologies., Comment: 19 pages, 12 figures, 6 tables. Accepted for publication in MNRAS