CatDat

finitary algebraic

We call a category finitary algebraic if it is equivalent to the category of models of a many-sorted finitary algebraic theory. There are several equivalent conditions:

1. It is equivalent to the category of finite-product-preserving $\L \to \Set$ from a small category $\L$ with finite products (=Lawvere theory).
2. It is equivalent to the category of models of a small finite-product sketch.
3. It is equivalent to the Eilenberg–Moore category of a finitary (=filtered-colimit-preserving) monad on $\Set^S$ for some set $S$.
4. It is equivalent to the Eilenberg–Moore category of a sifted-colimit-preserving monad on $\Set^S$ for some set $S$. (cf. [KR12, Proposition 3.3])
5. It is locally strongly finitely presentable, i.e. it is cocomplete and there is a set $G$ of strongly finitely presentable objects such that every object is a sifted colimit of objects from $G$.

• Related properties: generalized variety, locally finitely presentable, multi-algebraic, one-sorted finitary algebraic
• nLab Link

Relevant implications

• finitary algebraic is equivalent to cocomplete andgeneralized variety
• finitary algebraic implies locally finitely presentable
• finitary algebraic implies multi-algebraic
• finitary algebraic implies regular
• one-sorted finitary algebraic implies finitary algebraic

Examples

There are 31 categories with this property.

• category of abelian groups
• category of algebras
• category of cochain complexes of abelian groups
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of graded abelian groups
• category of graded modules over a graded ring
• category of groups
• category of Jónsson-Tarski algebras
• category of left modules over a division ring
• category of left modules over a non-semisimple ring
• category of left modules over a ring
• category of M-sets
• category of monoids
• category of pairs of sets
• category of pointed sets
• category of rings
• category of rngs
• category of semigroups
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of sets
• category of simplicial sets
• category of vector spaces
• partially ordered set of extended natural numbers
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 64 categories without this property.

• category of Banach spaces with linear contractions
• category of combinatorial species
• category of compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of fields
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finite-dimensional vector spaces
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• category of finitely generated abelian groups
• category of free abelian groups
• category of Hausdorff spaces
• category of locally ringed spaces
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• category of non-empty sets
• category of partially ordered sets
• category of pointed topological spaces
• category of preordered sets
• category of pseudo-metric spaces with non-expansive maps
• category of schemes
• category of sets and relations
• category of sets with a distinguished subset
• category of sets with finite-to-one maps
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of Z-functors
• delooping of a group
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• dual of the category of sets
• dual of the category of topological spaces
• empty category
• forked commutative square
• partially ordered collection of ordinal numbers
• partially ordered set of natural numbers
• preordered set of integers w.r.t. divisibility
• real interval [0,1]
• simplex category
• walking coreflexive pair
• walking fork
• walking idempotent
• walking parallel pair
• walking span
• walking splitting

Unknown

There are 2 categories for which the database has no information on whether they satisfy this property. Please help us fill in the gaps by contributing to this project.

• category of abelian sheaves
• category of sheaves