walking_pair.yaml

id: walking_pair
name: walking parallel pair
notation: $\Pair$
objects: two objects $0$ and $1$
morphisms: the two identities and two parallel morphisms from $0$ to $1$
description: >-
  This category can be pictured as:
  $$\{0 \rightrightarrows 1 \}$$
  Its name comes from the fact that a functor $\Pair \to \C$ is the same as a parallel pair of morphisms in $\C$.
nlab_link: https://ncatlab.org/nlab/show/walking+structure
tags:
  - category theory

related:
  - walking_coreflexive_pair
  - walking_fork
  - walking_morphism
  - forked_commutative_square

satisfied_properties:
  - property: small
    proof: This is trivial.

  - property: finite
    proof: This is trivial.

  - property: self-dual
    proof: This is trivial.

  - property: semi-strongly connected
    proof: This is trivial.

  - property: skeletal
    proof: The two objects are not isomorphic.

  - property: one-way
    proof: This is trivial.

  - property: extremal generator
    proof: It is easy to check that $0$ is an extremal generator.

  - property: left cancellative
    proof: The two morphisms $0 \rightrightarrows 1$ are clearly monomorphisms.

unsatisfied_properties:
  - property: strongly connected
    proof: There is no morphism $1 \to 0$.

  - property: pullbacks
    proof: 'The two morphisms $a,b : 0 \rightrightarrows 1$ have no pullback, since it would have to consist of identities $0 \leftarrow 0 \rightarrow 0$, but $a \neq b$.'

special_objects: {}

special_morphisms: {}