nLab
locally finitely presentable category

An object X of a category C is said to be finitely presentable (or finitary, sometimes called compact) if the representable functor C(X,) preserves filtered colimits. Write C fp for the full subcategory of C on the finitely presentable objects.

Mike: Do people really call finitely presentable objects “finitary”? I’ve only seen that word applied to functors (those that preserve filtered colimits).

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 a finitary essentially algebraic theory.

    Does this essentially algebraic theory also have to be finitary?; that is, if it's an algebraic theory, then it's a Lawvere theory? —Toby

    Mike: Yes, it certainly has to be finitary. Possibly the standard meaning of “essentially algebraic” implies finitarity, though, I don’t know.

    Toby: I wouldn't use ‘algebraic’ that way; see algebraic theory.

  3. C is equivalent to the category of finite-limit-preserving functors DSet for some small category D with finite limits.

  4. C fp has finite colimits, and the restricted Yoneda embedding C[C fp op,Set] identifies C with the category of finite-limit-preserving functors C fp opSet.

  5. C is the category of models for a finite limit sketch.

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

Examples