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The ring-shaped cohesin complex topologically entraps two DNA molecules to establish sister chromatid cohesion. Cohesin also shapes the interphase chromatin landscape with wide-ranging implications for gene regulation, and cohesin is thought to achieve this by actively extruding DNA loops without topologically entrapping DNA. The 'loop extrusion' hypothesis finds motivation from in vitro observations—whether this process underlies in vivo chromatin loop formation remains untested. Here, using the budding yeast S. cerevisiae, we generate cohesin variants that have lost their ability to extrude DNA loops but retain their ability to topologically entrap DNA. Analysis of these variants suggests that in vivo chromatin loops form independently of loop extrusion. Instead, we find that transcription promotes loop formation, and acts as an extrinsic motor that expands these loops and defines their ultimate positions. Our results necessitate a re-evaluation of the loop extrusion hypothesis. We propose that cohesin, akin to sister chromatid cohesion establishment at replication forks, forms chromatin loops by DNA–DNA capture at places of transcription, thus unifying cohesin's two roles in chromosome segregation and interphase genome organisation. Synopsis: The loop extrusion hypothesis has shaped current thinking about chromatin organization by SMC complexes. This article experimentally tests the hypothesis, with a negative outcome, and puts forward transcription as an alternative, external motor force promoting chromatin loop formation. Yeast cohesin variants unable to perform in vitro DNA loop extrusion readily form chromatin loops in vivo. Transcription occupies key roles in chromatin loop formation and growth. In a unified model, cohesin sequentially captures two DNAs to establish sister chromatid cohesion at DNA replication forks and chromatin loops at places of transcription. Cohesin variants defective in DNA loop extrusion in vitro can still form chromatin loops in living yeast cells. [ABSTRACT FROM AUTHOR]