In a Lawvere theory, one specifies an algebraic theory by giving a small category $T$ with finite products. Then a model of the theory is given by a functor $T \to Set$ that preserves products.
A sketch is a generalization of this idea, where we take a small category $T$, and pick some distinguished limit and colimit cones. Then a model of the sketch is a functor that preserves these limits and colimits.
A sketch is a small category $T$ equipped with a set $L$ of cones and a set $C$ of cocones. A realized sketch is one where all the cones in $L$ are limit cones and all the cocones in $C$ are colimit cocones.
A limit-sketch is a sketch with $C = \emptyset$, while a colimit-sketch is a sketch with $L = \emptyset$.
A model of a sketch in a category $\mathcal{C}$ is a functor $T\to \mathcal{C}$ taking each cone in $L$ to a limit cone and each cocone in $C$ to a colimit cocone. In particular, $T$ is realized if and only if its identity functor is a model. Frequently the notion of model is restricted to the case $\mathcal{C}=Set$.
A category is called sketchable if it is the category of models (in $Set$) of a sketch.
A Lawvere theory is a special case of a (limit-)sketch, where the category is one with a distinguished object $X$ such that all objects are (isomorphic to) powers of $X$, and $C = \emptyset$ and $L$ is the set of all product cones.
The categories of models of sketches are equivalently the accessible categories.
The categories of models of limit-sketches are the locally presentable categories.
From the discussion there we have that
an accessible category is equivalently:
a locally presentable category is equivalently:
We can “break in half” the difference between the two and define
and
The category of sketches is well behaved: it is complete, cocomplete, cartesian closed and has a second symmetric monoidal closed structure.
The category of sketches is topological over the category of directed pseudographs.
The above proposition gives the category of sketches Cartesian products - however these are often not the sketches one would expect when thinking of the product of two theories. Instead consider the tensor product:
Let $S,T$ be sketches. We define the sketch $S \otimes T$ to be:
The vertices of $S \otimes T$ are the product of the set of vertices from $S$, $T$. The set of arrows is given as
where the source of $(\alpha,b)$ is $(s_S(\alpha), b)$, and vice versa. $S$ is often called the horizontal structure and $T$ as the vertical structure. The set of diagrams is the union of the following three sets:
The set of cones and cocones are define analogously to the set of commuting diagrams, except only the vertical and horizontal cones are taken.
This tensor product, along with the unit $(*, \emptyset, \emptyset, \emptyset)$, gives the category of sketches a second symmetric monoidal category.
This monoidal structure is useful for considering structures like double categories (i.e. categories in the category of categories).
Let $S,T$ be sketches, and $X$ some category. Then the category of models of $S$ in the category of models of $T$ in $X$ is equivalent to the category of models of $S \otimes T$ in $X$.
An overview of the theory is given in
An extensive treatment of the links between theories, sketches and models can be found in
Society, Rhode Island, 1989.
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
The category of sketches itself was studied as a semantics for type theory in:
Last revised on March 25, 2019 at 18:41:54. See the history of this page for a list of all contributions to it.