Hochschildlike

Hochschildlike is a descriptive term used in mathematics to refer to structures, invariants, or theories whose formal properties resemble those of Hochschild cohomology and Hochschild homology. It is not a fixed, universally defined notion; its exact meaning varies by author and context.

Typically, Hochschildlike constructions involve a graded vector space or chain complex equipped with a differential, and they exhibit functoriality with respect to algebra maps or morphisms. They often carry algebraic structures such as a cup product or a Gerstenhaber bracket in cohomology, and may satisfy formal properties analogous to Morita invariance, excision, or long exact sequences.

Hochschildlike ideas appear in various settings, including associative algebras, differential graded (DG) algebras, A-infinity algebras, and DG categories. They also extend to geometric and topological contexts through invariants like Hochschild (co)homology of categories and their topological analogues. Topological Hochschild homology (THH) and cyclic or periodic cyclic homology are sometimes described as Hochschildlike in broader discussions, since they generalize the Hochschild theory to spectra or noncommutative geometries.

Usage and distinction: labeling a construction as Hochschildlike signals that it plays a Hochschild-type role without being literally Hochschild cohomology or homology. When a precise meaning is required, authors specify the exact construction (for example, the Hochschild complex HH^*(A,M) or THH(A)) and discuss which properties carry over. The term is more common in expository or comparative contexts than in a formal, universally adopted framework.