locally finitely presentable category

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A **locally finitely presentable category** is an ℵ${}_0$-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**):

- $C$ 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.
- $C$ 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 $D \to Set$.
- $C$ is the category of models for a finite limit sketch.
- $C_{fp}$ 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.

Last revised on October 15, 2012 at 17:52:44. See the history of this page for a list of all contributions to it.