nLab bi-

Meanings

In category theory, the prefix “bi-” is used in several ways with different (and not always compatible) meanings.

Weak 2-dimensional

Here, “bi-” is used to denote a weak 2-dimensional analogue of a 1-categorical concept.

Examples:

• bicategory
• biadjunction
• bilimit

In some cases, the prefix “pseudo-” may be used instead, such as pseudoadjunction rather than biadjunction. However, be warned that in some cases, these (unfortunately) refer to different concepts: e.g. a pseudolimit is a bilimit, but the converse is not always true.

X and also co-X

Here, “bi-X” is used to denote a structure that is equipped with both X-structure and dual co-X-structure (for some definition X) that do not necessarily coincide.

Examples:

• bicartesian category

• bireflective subcategory

• bifibrant objects

• bifibration

An unambiguous alternative is to simply be explicit about both structures being assumed.

Compatible X and co-X

Here, “bi-X” is used to denote a structure that is equipped with X-structure and co-X structure (not coincident) that verify some compatibility.

Examples:

• bimonoid
• bimonad

In many cases, these concepts have different, unambiguous names (such as Hopf monad) that may be used instead.

Coincident X and co-X

Here, “bi-X” is used to denote a structure that is equipped with X-structure and co-X-structure that coincides. This is a particularly common special case of the meaning above.

Examples:

• bilimit
• biproduct

An alternative, unambiguous prefix that can be used for this meaning is “ambi-”.

Left-X and also right-X

In some settings without symmetry, “bi-X” is occasionally used to denote having X-structure on the left and also X-structure on the right (not coincident).

Examples:

• biclosed
• bimodule
• bicomodule

In such settings, it is unambiguous to simply elide the prefix “bi-”. E.g. one can simply write “closed” to mean left- and right-closed.

Note that this is not the same as “X and co-X” (except in the delooping): a “biclosed category” with this terminology would be one that is closed and coclosed?.

Some compatibility may be required between the left and right X-structures.

Two compatible X structures

A less common usage is the prefix “bi-” to mean two structures of the same kind that interact in some way.

Examples:

• bimonoidal category (i.e. rig category)

However, these examples tend to have alternative, more descriptive terminology.

Related pages

• locally

category: disambiguation