We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2category into 2Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.

IntroductionWhile Cumulus – a semi-automated method for measuring breast density – is utilised extensively in research, it is labour-intensive and unsuitable for screening programmes that require an efficient and valid measure on which to base screening recommendations. We develop an automated method to measure breast density (AutoDensity) and compare it to Cumulus in terms of association with breast cancer risk and breast cancer screening outcomes.MethodsAutoDensity automatically identifies the breast area in the mammogram and classifies breast density in a similar way to Cumulus, through a fast, stand-alone Windows or Linux program. Our sample comprised 985 women with screen-detected cancers, 367 women with interval cancers and 4,975 controls (women who did not have cancer), sampled from first and subsequent screening rounds of a film mammography screening programme. To test the validity of AutoDensity, we compared the effect estimates using AutoDensity with those using Cumulus from logistic regression models that tested the association between breast density and breast cancer risk, risk of small and large screen-detected cancers and interval cancers, and screening programme sensitivity (the proportion of cancers that are screen-detected). As a secondary analysis, we report on correlation between AutoDensity and Cumulus measures.ResultsAutoDensity performed similarly to Cumulus in all associations tested. For example, using AutoDensity, the odds ratios for women in the highest decile of breast density compared to women in the lowest quintile for invasive breast cancer, interval cancers, large and small screen-detected cancers were 3.2 (95% CI 2.5 to 4.1), 4.7 (95% CI 3.0 to 7.4), 6.4 (95% CI 3.7 to 11.1) and 2.2 (95% CI 1.6 to 3.0) respectively. For Cumulus the corresponding odds ratios were: 2.4 (95% CI 1.9 to 3.1), 4.1 (95% CI 2.6 to 6.3), 6.6 (95% CI 3.7 to 11.7) and 1.3 (95% CI 0.9 to 1.8). Correlation between Cumulus and AutoDensity measures was 0.63 (P < 0.001).ConclusionsBased on the similarity of the effect estimates for AutoDensity and Cumulus in models of breast density and breast cancer risk and screening outcomes, we conclude that AutoDensity is a valid automated method for measuring breast density from digitised film mammograms.

The Catalan simplicial set C is known to classify skew-monoidal categories in the sense that a map from C to a suitably defined nerve of Cat is precisely a skew-monoidal category [1]. We extend this result to the case of skew monoidales internal to any monoidal bicategory B. We then show that monoidal bicategories themselves are classified by maps from C to a suitably defined nerve of Bicat and extend this result to obtain a definition of skew-monoidal bicategory that aligns with existing theory.

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