nLab bilimit

The word bilimit is used in two unrelated senses:

• 2-categorical limits – In the context of bicategory theory, bilimit is the relevant notion of categorical limit. If one adopts the convention that 2-categories are understood in the most general sense and subsume bicategories, then these are just called 2-limits. See there for more on this sense of the word.

• limits coinciding with colimits – It is common to speak of biproducts to mean categorical products that coincide with coproducts. Since a product is a special kind of limit, this terminology conflicts with the above one. Indeed, some references use “bilimit” to mean limits that coincide with colimits (e.g. Selinger 97 (pdf)).

  In a category enriched over CMon (in particular over Ab) biproducts are a special case of absolute colimits, see at absolute colimit – Examples – Weights for absolute colimits.

  More generally, since limits and colimits are a special case of right adjoints and left adjoints, respectively (namely left and right Kan extension to the point), bilimits in this sense are a special case of an ambidextrous adjunction, which is therefore maybe a preferable terminology (proposed this way in Hopkins-Lurie 14).

  Notice however that being an ambidextrous adjunction is structure, since in general it involves a genuine choice of isomorphism between the left and the right adjoint, whereas bilimits in the above sense are typically regarded as property. On the other hand, at least for (co-)products the existence of any natural isomorphism between coproducts and products already implies that also the canonical comparison morphism is an equivalence, which is traditionally taken to be the property that characterizes semiadditive categories, see at biproduct – Properties – Semiadditivity as structure/property. Similarly, one may strengthen the notion of ambidextrous adjunction such as to make it canonical, see at fiberwise characterization.

category: disambiguation