I encountered this term in [1]. I can’t find a formal definition. Here is the definition from [1, p. 23]:

We shall call a category algebraic if it is monadic over Set, and equationally presentable if its objects can be described by (a proper class of) operations and equations…

The term “described” has no formal definition. From [2, p. 334]:

Recall that a category is called algebraic if it is monadic over Set, and equationally presentable if its objects can be prescribed by (a proper class of) operations and equations.

Again, no definition of “prescribed”. I suppose that the definition of an equational class of algebras from universal algebra does not fit because it is not about categories. BTW, what are “operations” of a frame? The join operation accepts a set of arguments, so it is not a usual algebraic operation.

  1. Johnstone, Peter T. Stone Spaces. Cambridge: Cambridge UP, 1986. Print.

  2. Kruml, D, and J. Paseka. “Algebraic and Categorical Aspects of Quantales.” Handbook of Algebra. Ed. M. Hazewinkel. Vol. 5. Amsterdam: Elsevier, 2008. 323–62. Print.

  • $\begingroup$ Of course universal algebra can be about categories. If you have a variety $K$ of algebras, then there is a category whose objects are the elements of $K$ and whose morphisms are structure-preserving maps between these elements. I think that is what's meant here $\endgroup$ – Max Apr 4 '18 at 13:34

The categories being discussed are, conceptually, the categories of models for an algebraic theory a la universal algebra. If I'm reading you right, the class of objects of this category will be the "equational class of algebras". The arrows are homomorphisms of algebras as Max suggests.

So "equationally presented" means equationally presented in the same sense of universal algebra. For a categorist, this is usually taken to mean that the category is a category of models for an algebraic theory. However, categorists often go beyond the usual constraints of universal algebra allowing potentially a proper class of operations. Also, the arity of the operations is allowed to be any arity in a given arity class. As an extreme example of an arity class, we can allow any cardinal arity in which case we call the theory infinitary. We'll often require the theory to be locally small meaning for each arity there is a set of operations, not a proper class. For this, the algebraic theory of frames is infinitary but locally small.

Basically, the categorical approach to traditional (single-sorted) universal algebra is to use a Lawvere theory which is a small category containing exactly all finite products of a single object. (Set-theoretic) Models of a Lawvere theory are then finite-product preserving functors from the Lawvere theory into $\mathbf{Set}$. This corresponds almost exactly to traditional (single-sorted) universal algebra. The only issue is a Lawvere theory effectively "forgets" its presentation. It's conceptually straightforward to consider arbitrary (small) products and arbitrary product preserving functors instead. This will lead to a proper class of objects but we can still require the theory to be locally small. Being a category of models for such a "large" Lawvere theory turns out to be equivalent to having a monad on $\mathbf{Set}$. However, we can have situations like complete Boolean algebras where the corresponding Lawvere theory is not even locally small, at which point a monad does not exist. As a simpler example, consider an infinitary signature with one operation, $u_\kappa$, for each arity, i.e. for each cardinal $\kappa$, and no equations. Then there is a proper class of terms with one free variable, namely $u_\kappa(\_\mapsto x)$ for each $\kappa$ which are all distinct because there are no equations. So if we allow a proper class of operations and equations, an equationally presented category isn't necessarily representable as a locally small Lawvere theory nor monadic.

  • $\begingroup$ I would like to hear a formal definition, finally. $\endgroup$ – beroal Apr 8 '18 at 17:54
  • $\begingroup$ I gave a categorical definiton, models of a large Lawvere theory, i.e. product preserving functors from a large Lawvere theory. In universal algebra terms, you basically just replace "set" by "class" everywhere. You have an arity class $A$, a class of operations $O$, a class function $\mathsf{ar}:O\to A$ assigning arities to the operations. Given a set $V$, a term with variables from $V$ is a pair of a subset $S\subset O$ and a term with variables from $V$ and operations in $S$ in the usual sense. Equations are pairs of terms. $\endgroup$ – Derek Elkins Apr 8 '18 at 23:21
  • $\begingroup$ The set of $n$-tuples on $X$, $X^n$ can be viewed as the set of functions $[n]\to X$ where $[n]$ is a set of cardinality $n$. This generalizes to any cardinality. Given $f\in S$ with arity $\kappa$ and a function, $\bar x$, from $\kappa$ to terms with operations in $S$, $f(\bar x)$ is a term. An interpretation is a set $X$ and a class indexed family of functions $\{f_o:X^{\mathsf{ar}(o)}\to X\}_{o\in O}$ whose action on terms is defined recursively in the usual way and which validates all equalities in the usual way. Homomorphism is defined in the usual way. $\endgroup$ – Derek Elkins Apr 8 '18 at 23:21

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