Implication Details

Claim: If a category has a generating set and is locally essentially small, then it is concretizable.

Proof: If $S$ is a generating set of a locally small category $\C$ , then by definition the functor $(\Hom(G,-))_{G \in S} : \C \to \Set^S$ is faithful. Furthermore, the functor $\Set^S \to \Set$ mapping $X \mapsto \coprod_{G \in S} X_G$ is faithful. Their composition provides a faithful functor $\C \to \Set$ .

Show 97 categories using this implication

empty category
trivial category
discrete category on two objects
category of abelian groups
category of finitely generated abelian groups
category of algebras
category of finite sets and bijections
delooping of a group
delooping of an infinite countable group
delooping of a non-trivial finite group
delooping of an infinite uncountable group
delooping of the additive monoid of natural numbers
category of commutative algebras
category of commutative monoids
category of commutative rings
category of small categories
category of cochain complexes of abelian groups
category of compact Hausdorff spaces
simplex category
category of coproducts of Euclidean spaces
category of finite sets and injections
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 of even cardinality
category of finite sets of odd cardinality
category of finite sets of cardinality a power of 3
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 fields
category of fields of characteristic zero
category of finitely generated free abelian groups
category of finitely generated free modules over Z x Z
category of groups
category of countable groups
category of Hausdorff spaces
category of Jónsson-Tarski algebras
category of M-sets
category of smooth manifolds
category of measurable spaces
category of metric spaces with non-expansive maps
category of metric spaces with continuous maps
category of metric spaces with ∞ allowed
category of monoids
category of sets with a distinguished subset
partially ordered set of natural numbers
partially ordered set of extended natural numbers
partially ordered collection of ordinal numbers
category of pseudo-metric spaces with non-expansive maps
category of partially ordered sets
category of partially ordered sets without isolated points
category of preordered sets
category of finitely generated projective modules over the ring of dual numbers
category of left modules over a ring
category of left modules over a division ring
category of left modules over a non-semisimple ring
category of sets and relations
category of rings
category of rngs
category of semigroups
category of sequences of abelian groups
category of sets
category of set functions and commutative squares
category of countable sets
discrete category of sets
category of sets with finite-to-one maps
category of non-empty sets
category of pairs of sets
category of sheaves
category of abelian sheaves
category of topological spaces
category of pointed topological spaces
category of torsion abelian groups
category of uniform spaces
category of vector spaces
category of countable-dimensional vector spaces
preordered set of integers w.r.t. divisibility
cocompletion of a discrete–pair join
forked commutative square
category of graded abelian groups
category of graded modules over a graded ring
real interval [0,1]
category of simplicial sets
walking commutative square
walking composable pair
walking coreflexive pair
walking fork
walking idempotent
walking isomorphism
walking morphism
walking parallel pair
walking span
walking splitting