nLab
sketch

Context

Category theory

Limits and colimits

Contents

Idea

In a Lawvere theory, one specifies an algebraic theory by giving a small category TT with finite products. Then a model of the theory is given by a functor TSetT \to Set that preserves products. A sketch is a generalization of this idea, where we take a small category TT, and pick some distinguished limit and colimit cones. Then a model of the sketch is a functor that preserves these limits and colimits.

Definition

Definition

A sketch is a small category TT equipped with subsets (L,C)(L, C) of its limit cones and colimit cocones.

A limit-sketch is a sketch with just limits and no colimits specified, ie. with C=C = \emptyset. Dually, a colimit-sketch is a sketch with L=L = \emptyset.

A model of a sketch is a Set-valued functor preserving the specified limits and colimits.

A category is called sketchable if it is the category of models of a sketch.

Examples

Example

A Lawvere theory is a special case of a (limit-)sketch, where the category is one with a distinguished object XX such that all objects are (isomorphic to) powers of XX, and C=C = \emptyset and LL is the set of all product cones.

Properties

Relation to accessible and locally representable categories

Proposition

The categories of models of sketches are equivalently the accessible categories.

Proposition

The categories of models of limit-sketches are the locally presentable categories.

Remark

From the discussion there we have that

  • an accessible category is equivalently:

    • a full subcategory of a presheaf category that’s closed under κ\kappa-filtered colimits for some κ\kappa
    • the category of models of a sketch
  • a locally presentable category is equivalently:

    • a reflective full subcategory of a presheaf category that’s closed under κ\kappa-filtered colimits for some κ\kappa
    • the category of models of a limit sketch
    • an accessible category with all small limits
    • an accessible category with all small colimits

We can “break in half” the difference between the two and define

  • a locally multipresentable category to be equivalently:
    • a multireflective full subcategory of a presheaf category that’s closed under κ\kappa-filtered colimits for some κ\kappa
    • the category of models of a limit and coproduct sketch
    • an accessible category with all small connected limits
    • an accessible category with all small multicolimits

and

  • a weakly locally presentable category to be equivalently:
    • a weakly reflective full subcategory of a presheaf category that’s closed under κ\kappa-filtered colimits for some κ\kappa
    • the category of models of a limit and epi sketch
    • an accessible category with all small products
    • an accessible category with all small weak colimits

References

An overview of the theory is given in

An extensive treatment of the links between theories, sketches and models can be found in

  • Michael Makkai, Robert Paré, Accessible categories: The foundations of categorical model theory Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.

That not only every sketchable category is accessible but that conversely every accessible category is sketchable is due to

  • Christian Lair, Catégories modelables et catégories esquissables, Diagrammes (1981).

Revised on August 29, 2016 04:44:39 by Dexter Chua (113.255.131.223)