Auditory categorization is a crucial aspect of everyday communication, but we lack a clear understanding of the computations performed by the brain to define boundaries between groups of natural sounds. While modern machine-learning-based classifiers perform as well as humans in categorizing sounds, they can rely on idiosyncratic acoustic cues that diverge from biologically relevant cues. We hypothesized that, since auditory cortical neurons show robust category selectivity, training a classifier with an auditory cortical front end should more accurately reflect biological categorization. To test this, we trained an encoding model to predict single-unit spike train data from the ferret auditory cortex. To capture the immense heterogeneity in auditory cortical response properties and natural soundscapes, we used a multi-task architecture to train a single convolutional-neural-network-based encoding model across multiple neural recordings, each collected with different natural sound stimuli. The bottleneck layer of this comprehensive model yielded the "auditory cortical manifold," which accounted for the spectrotemporal features driving responses of the entire cortical population. Representational similarity analysis showed that the manifold was consistent across animals and thus captured general cortical computations for natural sound processing. We pooled data across animals to train a single encoding model and used the manifold representation to train a classifier. Additionally, we trained a spectrogram-based classifier (without neural grounding) and a neural classifier (using the spike count of auditory cortical neurons) on the same categorization task. Confusion patterns for the manifold-based classifier were better aligned with the neural classifier than with the spectrogram classifier, underscoring the value of using neural data to develop biologically grounded classifiers. These findings emphasize the potential of computational models that mimic sensory processing to achieve more biologically relevant performance.