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Chapter 1
Unital categories and intrinsic commutation
1.1 Notation
Let E be a finitely complete category. Given any map f : X → Y , we denote the kernel equivalence relation of this map by R[f]. Given the following right hand side commutative square, we denote by R(x) the map induced by the map x between the respective kernel equivalences:
p1 R(x)
p0
// f R[f]oo s0 // X // Y
′ oo p′0 R[f]s′0 //X′
p′ 1
See the Appendix for further details.
1.2 Unital categories
//Y′.
//
xy
f′
In this section, C will be a pointed category, i.e. a category with a zero object 0 (in the category Mon of monoids, it is given by the trivial monoid with only one element). Let us recall from [8]:
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