# nLab PIE-limit

Contents

### Context

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

#### Limits and colimits

limits and colimits

# Contents

## Idea

In 2-category-theory, by a PIE-limit one means a strict 2-limit which can be constructed from

1. (P) strict products,

2. (I) strict inserters,

3. (E) strict equifiers.

More precisely, the class of PIE-limits is the saturation of the class containing products, inserters, and equifiers. Any PIE-limit is in particular a flexible limit, and therefore also a (non-strict) 2-limit.

Furthermore, all strict pseudo-limits are PIE-limits, and therefore any strict 2-category which admits all PIE-limits also admits all non-strict 2-limits, although it may not have all strict 2-limits. This is the case, for instance, for the 2-category of strict algebras and pseudo morphisms over a strict 2-monad.

Some examples of PIE-limits are:

An intuition is that PIE-limits are those 2-dimensional limits that do not impose any equations between 1-cells. For instance, equalizers and pullbacks are not PIE-limits.

PIE-limits can also be characterized as the coalgebras for a pseudo morphism classifier? comonad, exhibiting them as a 2-categorical version of the notion of rigged limit.

## Characterisation of PIE-weights

A PIE-limit is one whose weight is PIE. Power and Robinson characterised such weights $W : J^{op} \to Cat$ as those for which the induced functor $ob \circ W_0 \colon J^{op}_0 \to Cat_0 \to Set$ is multirepresentable. This holds if and only if each connected component of the category of elements of $ob \circ W_0$ has an initial object.

## References

On $\lambda$-small objects in PIE-limits:

Last revised on June 12, 2024 at 09:35:43. See the history of this page for a list of all contributions to it.