locally finitely presentable category



Category theory

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Compact objects

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A locally finitely presentable category is an 0{}_0-locally presentable category.

We spell out what this means:

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

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

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

Replacing “finite” by “of cardinality less than κ\kappa” everywhere, for some cardinal number κ\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.