nLab
sketch

Definition

Definition

A sketch is a small category equipped with a subset of its limit cones and colimit cocones.

A limit-sketch is a sketch with just limits and no colimits specified.

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.

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 $\kappa$-filtered colimits for some $\kappa$
  • 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 $\kappa$-filtered colimits for some $\kappa$
  • 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 $\kappa$-filtered colimits for some $\kappa$
  • 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 $\kappa$-filtered colimits for some $\kappa$
  • 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

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)

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, 1989.1989.

• Michael Barr, Charles Wells, Toposes, Triples, and Theories, Originally published by: Springer-Verlag, New York, 1985, republished in: Reprints in Theory and Applications of Categories, No. 12 (2005) pp. 1-287

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).