Cerebral cortex has been hypothesized to operate close to a critical phase transition. This hypothesis offers an explanation of the observed complexity of brain dynamics and is important because of potential computational advantages near criticality. However, in the awake state, when cortex most needs computation, experimental evidence for criticality has been inconsistent, especially when considering high precision measurements, i.e. spikes of many single neurons measured with millisecond resolution. The inconsistency of previous reports casts doubt on the possibility that awake cortex operates near criticality. Here we show that discrepant previous reports of critical phenomena in the brain may be reconciled by considering dimensionality and dimensionality reduction of brain dynamics. Indeed, fundamental physics of critical phenomena emphasizes the importance of coarse-graining of observables, which is a type of dimensionality reduction. Many detailed microscopic degrees of freedom must be excluded to reveal universal macroscopic features of criticality. We show that coarse graining over neurons and time is a type of dimensionality reduction which reveals low-dimensional critical dynamics in a prominent subspace (first few principal components) of awake cortical dynamics.