Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
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.
Relation to accessible and locally representable categories
From the discussion there we have that
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 -filtered colimits for some
- 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
- a weakly locally presentable category to be equivalently:
- a weakly reflective full subcategory of a presheaf category that’s closed under -filtered colimits for some
- 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
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, 1989.1989.
That not only every sketchable category is accessible but that conversely every accessible category is sketchable is due to
Revised on March 6, 2014 08:33:04
by Tim Porter