# nLab locally finitely presentable category

Contents

### Context

category theory

#### Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

# Contents

## Definition

###### Definition

A locally finitely presentable category is an ${}_0$-locally presentable category.

We spell out what this means:

An object $X$ of a category $C$ is said to be finitely presentable (sometimes called compact or ‘finite’) if the representable functor $C(X,-)$ is finitary, i.e., preserves filtered colimits. Write $C_{fp}$ for the full subcategory of $C$ consisting of the finitely presentable objects.

A category $C$ satisfying (any of) the following equivalent conditions is said to be locally finitely presentable (or lfp):

1. $C$ has all small colimits, the category $C_{fp}$ is essentially small, and any object in $C$ is a filtered colimit of the canonical diagram of finitely presentable objects mapping into it.
2. $C$ is the category of models for an essentially algebraic theory. Here an ‘essentially algebraic theory’ is a small category $D$ with finite limits, and its category of ‘models’ is the category of finite-limit-preserving functors $D \to Set$. (See Gabriel–Ulmer duality.)
3. $C$ is the category of models for a finite limit sketch.
4. $C_{fp}$ has finite colimits, and the restricted Yoneda embedding $C\hookrightarrow [C_{fp}^{op},Set]$ identifies $C$ with the category of finite-limit-preserving functors $C_{fp}^{op} \to Set$.

Replacing “finite” by “of cardinality less than $\kappa$” everywhere, for some cardinal number $\kappa$, results in the notion of a locally presentable category.

## Examples

• P. Gabriel, F. Ulmer, Lokal präsentierbare Kategorien, Springer Lect. Notes in Math. 221 1971 Zbl0225.18004 MR327863
• Jiří Adámek, Jiří Rosicky, Locally presentable and accessible categories, Cambridge University Press 1994.

Introductory account:

If $V$ is a locally finitely presentable symmetric monoidal closed category then there is a bijection between exact $V$-localizations of the $V$-category of $V$-valued $V$-enriched presheaves on a $V$-category $C$ and $V$-enriched Grothendieck topologies on $C$: