CatDat

essentially small

A category is essentially small when it is equivalent to a small category. In particular, there is a set of objects such that every object is isomorphic to an object in this set. In contrast to the property of being small, being essentially small is invariant under equivalences of categories.

• Dual property: essentially small (self-dual)
• Related properties: small
• nLab Link

Relevant implications

• small implies   locally small and  essentially small

  trivial
• essentially small implies   well-powered and  well-copowered and  locally essentially small

  trivial
• essentially small and  products implies   thin

  Mac Lane, V.2, Prop. 3. The proof works for any category with products.
• essentially finite implies   essentially small

  trivial
• essentially small and  thin and  complete implies   cocomplete

  The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.
• essentially small and  thin and  complete and  infinitary distributive implies   cartesian closed

  This is an application of the adjoint functor theorem. Specifically, if $P$ is a complete lattice in which $\sup_i \inf(t,x_i) = \inf(t, \sup_i y_i)$ always holds, then the functor $\int(t,-)$ is a left adjoint because it preserves all suprema.
• essentially small and  coproducts implies   thin

  [dualized] Mac Lane, V.2, Prop. 3. The proof works for any category with products.
• essentially small and  thin and  cocomplete implies   complete

  [dualized] The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.

Examples

• category of combinatorial species
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• discrete category on two objects
• empty category
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism
• walking parallel pair of morphisms

Counterexamples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of commutative rings
• category of fields
• category of free abelian groups
• category of groups
• category of left R-modules
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of vector spaces
• category of Z-functors
• delooping of the additive monoid of ordinal numbers
• partial order of ordinal numbers

Unknown

For these categories the database has no info if they satisfy this property or not.

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