CatDat

essentially injective

A functor $F : \C \to \D$ is essentially injective if the implication $$F(A) \cong F(B) \implies A \cong B$$ holds for all objects $A,B \in \C$. This is a condition solely on the objects themselves, i.e. it is not required that every isomorphism between $F(A)$ and $F(B)$ is induced by an isomorphism between $A$ and $B$ (cf. full on isomorphisms). An equivalent condition is that $F$ induces an injective map on isomorphism classes.

• Dual property: essentially injective (self-dual)
• Related properties: conservative, essentially surjective, full on isomorphisms, fully faithful, pseudomonic
• nLab Link

Relevant implications

• full on isomorphisms implies essentially injective
• left-invertible implies essentially injective
• () implies essentially injective

Examples

There are 27 functors with this property.

• binary diagonal functor on the category of sets
• discrete topology functor
• doubling functor on sets
• empty functor to the category of sets
• forgetful functor from abelian groups to groups
• forgetful functor from commutative rings to rings
• forgetful functor from finite abelian groups to abelian groups
• forgetful functor from finite groups to groups
• forgetful functor from finite sets to sets
• forgetful functor from groups to monoids
• forgetful functor from Hausdorff spaces to topological spaces
• forgetful functor from torsion abelian groups to abelian groups
• forgetful functor from torsion-free abelian groups to abelian groups
• free group functor
• identity functor on the category of sets
• inclusion functor from extended natural numbers to ordinal numbers
• indiscrete topology functor
• morphism endpoints inclusion
• nerve functor
• opposite category functor
• opposite monoid functor
• span endpoints inclusion
• squaring functor on sets
• trivial functor from the delooping
• trivial functor from the walking idempotent
• walking isomorphism object inclusion
• walking morphism representation

Counterexamples

There are 27 functors without this property.

• abelianization functor for groups
• binary coproduct functor on sets
• binary product functor on sets
• Brauer group functor
• countable copower functor on sets
• enveloping group functor
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for topological spaces
• forgetful functor for vector spaces
• forgetful functor from groups to pointed sets
• forgetful functor from rings to monoids
• functor of continuous functions
• fundamental group functor
• group of units functor
• modulo p functor
• monoid ring functor
• p-torsion functor
• path components functor
• rational product functor
• ring idempotents functor
• sequences functor on sets
• simple-group probing functor
• Stone-Čech compactification functor
• torsion functor
• trivial functor from the category of groups
• trivial functor from the category of sets

Undecidable functors

There are 2 functors for which it cannot be decided if this property is satisfied or not.

• contravariant power set functor
• covariant power set functor

Unknown

There are 0 functors for which the database has no information on whether they satisfy this property.

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