We derive three equivalent necessary conditions for a small category to have homological dimension one, generalizing a result of Novikov. As a consequence, any small cancellative category of homological dimension one is embeddable in a groupoid.
Keywords: homological dimension, cohomological dimension, small category, cancellative category, cyclic system, crown, supported crown, DCC category, Malcev sequence, embeddability into a group
2010 MSC: 18G20, 20M50
Theory and Applications of Categories, Vol. 35, 2020, No. 5, pp 137-154.