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counterexamples in category theory

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Contents

Idea

In category theory, there’s often the sense that β€˜things just work’, and that details can be skipped as everything is well behaved. Unfortunately, this is not quite true.

The counterexamples here should serve as specimens of possible pittfalls in category theory, and give an idea of what could go wrong.

List

  1. Sending an object xβˆˆπ’žx \in \mathcal{C} of a category π’ž\mathcal{C} to its automorphism group Aut π’ž(x)Aut_{\mathcal{C}}(x) (or its endomorphism monoid End π’ž(x)End_{\mathcal{C}}(x) ) does not in general extend to a functor from π’ž\mathcal{C} to Groups.

    It does however extend to a functor on the core of π’ž\mathcal{C} (the maximal groupoid inside it, keeping only the isomorphisms of π’ž\mathcal{C}), where it sends morphisms (now constrained to be isomorphisms) to their conjugation action:

    Core(π’ž) ⟢Aut π’ž Groups x Aut π’ž(x) ≃↓ Ξ³ ↦ ↓ ad Ξ³:gβ†¦Ξ³βˆ˜g∘γ βˆ’1 y Aut π’ž(y). \array{ Core(\mathcal{C}) & \overset{ \;\;\; Aut_{\mathcal{C}} \;\;\; }{ \longrightarrow } & Groups \\ x & & Aut_{\mathcal{C}}(x) \\ {}^{{}_{\mathllap{ \simeq }}} \big\downarrow {}^{{}_{\mathrlap{ \gamma }}} & \mapsto & \big\downarrow {}^{{}_{\mathrlap{ ad_\gamma \colon g \mapsto \gamma \circ g \circ \gamma^{-1} }}} & \\ y & & Aut_{\mathcal{C}}(y) \,. }

    For example, if π’ž=Ξ  1(X)\mathcal{C} = \Pi_1(X) is the fundamental groupoid of a topological space (which thus coincides with its core already), then the automorphism groups of its objects x∈Xx \in X are the fundamental groups Ο€ 1(X,x)\pi_1(X,x) at these basepoints, which famously are functorial under conjugation by paths in XX.

  2. Forming the center of a group does not extend to a functor from Groups to AbelianGroups.

    It does extend to a functor on the core, though:

    Core(Groups) ⟢Center AbelianGroups. \array{ Core(Groups) & \overset{ Center }{ \longrightarrow } & AbelianGroups \,. }

    Notice that this example is really a special case of the previous one (forming automorphism groups), or rather of a 2-category theoretic version of it: The center of a group is the automorphism group in the endo-functor category of the identity functor on the one-object delooping-groupoid BG\mathbf{B}G of GG:

    Center(G)≃Aut End(BG)(id BG) Center(G) \;\simeq\; Aut_{{}_{End(\mathbf{B}G)}} \big( id_{\mathbf{B}G} \big)
  3. Composing a monadic functor with another monadic functor need not be monadic. For example, Torsion-free abelian groups are monadic over abelian groups, which are monadic over sets, but torsion-free abelian groups are not monadic over sets.

  4. Taking the skeleton of a monoidal category does not in general result in a strict monoidal category. The argument in the case of Set is given in this MO answer. In particular, one cannot in general replace a monoidal category with an equivalent category that is simultaneously strict monoidal and skeletal.

  5. The category of topological spaces and local homeomorphisms is locally cartesian closed but not cartesian closed since it does not have a terminal object.

  6. There are functors D:Affβ†’VectD:Aff\to Vect (taking the vector space of displacements) and A:Vectβ†’AffA:Vect \to Aff (taking the underlying affine space) between the categories of vector spaces and affine spaces, and we have D(A(V))β‰…VD(A(V)) \cong V for any V∈VectV\in Vect, and for any U∈AffU\in Aff there exists some isomorphism A(D(U))β‰…UA(D(U)) \cong U (after choosing a point in UU to serve as the origin), but the categories are not equivalent β€” the second isomorphisms cannot be chosen naturally, not even after restricting to the cores.

  7. The opposite of the category of commutative von Neumann algebras has a subobject classifier and it’s finitely complete, but is not a topos since it is not cartesian closed. See this MO question.

  8. There is a β€˜wrong right adjoint’ to the functor F:Petriβ†’CMCF : Petri \to CMC (as described in Petri nets#semantics) which sends a commutative monoidal category CC to the Petri net whose source and target are η∘s:Mor(C)β†’β„•[Ob(C)]\eta \circ s : Mor(C) \to \mathbb{N}[Ob(C)] and η∘t:Mor(C)β†’β„•[Ob(C)]\eta \circ t : Mor(C) \to \mathbb{N}[Ob(C)] where ss and tt are the source and target of CC and Ξ·\eta is the unit for the monad β„•[βˆ’]\mathbb{N}[-]. In words, CC is sent to the Petri net whose source and target are the source and target of CC composed with Ξ·\eta. The natural choice of isomorphism Hom(FA,B)β‰…Hom(A,GB)Hom(FA, B) \cong Hom(A, GB), where GG is the functor just described, turns out not to yield well-defined morphisms of Petri nets. See Remark 4.4 in Master 2020

  9. Let CC be a category without an initial object and let R:C→1R : C \to 1 be the unique functor from CC into the terminal category. Then RR preserves all limits but does not have a left adjoint because this left adjoint would have to send the unique object of 1 to an initial object.

  10. Consider the functor U:Groupβ†’SetU\colon Group \to Set that sends a group GG to the set ∏ i∈Ihom(S i,G)\prod_{i\in I} hom(S_i, G), where the groups S iS_i are simple for all i∈Ii\in I, II is the class of all ordinals, and the cardinality of S iS_i eventually becomes bigger than any fixed cardinal as ii increases. This is a limit-preserving functor between locally presentable categories. However, it does not have a left adjoint.

  11. Dense functors are not closed under composition. For example, Ξ” <2\Delta_{\lt 2} is dense in the simplex category Ξ”\Delta and Ξ”\Delta is dense in Cat\mathbf{Cat}, but Ξ” <2\Delta_{\lt 2} is not dense in Cat\mathbf{Cat}.

See also

References

The initial import of counterexamples in this entry was taken from this Zulip discussion.

Last revised on March 17, 2021 at 04:37:08. See the history of this page for a list of all contributions to it.