cototal

A locally small category $\C$ is called cototal if it satisfies one of the following equivalent conditions:

1. The contravariant Yoneda embedding $$y : \C^{\op} \to [\C, \Set]$$ has a left adjoint.
2. Every discrete opfibration $X : \I \to \C$ (with $\I$ not necessarily essentially small) whose fibers are bijective to sets has a limit in $\C$. (Recall that $X$ is a discrete opfibration if for every morphism $f : X_i \to Y$ in $\C$ there exists a unique $\alpha : i \to j$ in $\I$ such that $X_\alpha = f$.)
3. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $X \downarrow Y$ is bijective to a set. Then $X$ has a limit in $\C$.

Relevant implications

• cogenerating set andcomplete andlocally essentially small andwell-powered implies cototal
• cototal andself-dual implies total
• cototal implies cocomplete
• cototal implies complete
• cototal implies locally essentially small
• self-dual andtotal implies cototal

Examples

There are 41 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of Banach spaces with linear contractions
• category of cochain complexes of abelian groups
• category of compact Hausdorff spaces
• category of filtered vector spaces
• category of graded abelian groups
• category of graded modules over a graded ring
• 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 measurable spaces
• category of metric spaces with ∞ allowed
• category of pairs of sets
• category of partially ordered sets
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of sets
• category of sets with a distinguished subset
• category of sheaves
• category of simplicial sets
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• dual of the category of sets
• dual of the category of topological spaces
• partially ordered set of extended natural numbers
• preordered set of integers w.r.t. divisibility
• real interval [0,1]
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 69 categories without this property.

• category of algebras
• category of combinatorial species
• category of commutative algebras
• category of commutative rings
• category of coproducts of Euclidean spaces
• category of countable groups
• category of countable sets
• category of countable-dimensional vector spaces
• category of fields
• category of fields of characteristic zero
• 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 sets of cardinality a power of 3
• category of finite sets of even cardinality
• category of finite sets of odd cardinality
• 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 finitely generated free abelian groups
• category of finitely generated free modules over Z x Z
• category of finitely generated projective modules over the ring of dual numbers
• category of free abelian groups
• category of groups
• category of Hausdorff spaces
• category of large vector spaces over a large field with a small basis
• category of long transfinite sequences of abelian groups
• 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 partially ordered sets without isolated points
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of set-indexed families of abelian groups
• category of sets and relations
• category of sets with finite-to-one maps
• category of small categories
• category of smooth manifolds
• category of Z-functors
• cocompletion of a discrete–pair join
• 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 of sets
• discrete category on two objects
• empty category
• forked commutative square
• partially ordered collection of ordinal numbers
• partially ordered set of natural numbers
• 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 commutative monoids
• category of locally ringed spaces