# Universal with respect to property

Reading Weibl's Introduction to Homological Algebra we get the definition

A kernel of a morphism $$f:B \to C$$ is defined to be a map $$i:A \to B$$ such that $$fi = 0$$ and that is universal with respect to this property.

I am trying to digest what it means to be universal with respect to a property. From what I understand this means that for any other object satisfying such a property there is a unique isomorphism between the two universal objects. Futher, an object is universal if it is either initial or terminal (and the kernel is clearly an initial morphism with A an initial object?) and the universal property is the universal morphism, that is, the initial or terminal morphism, (which in our case is $$i$$ which is an initial morphism?). It is my understanding that universal objects and universal properties(/morphisms?) occur together?

What is a universal property?

Being universal with respect to a property

What does "universal w.r.t. this property" mean? (kernel of a morphism in an additive category)

The second is a direct response to the same question but I don't quite find the answer complete enough for my understanding.

• A while back I was reading in Bergmann's Invitation to Universal Algebra, online, and there was a certain amount of this sort of thing. It was a good read. – Chris Custer Sep 8 '19 at 6:14

If the original category is $$\mathcal C$$, consider the category $$\mathcal D$$ whose objects are $$\mathcal C$$-morphism $$i\colon A\to B$$ with codomain $$B$$ and such that $$f\circ i=0$$ in $$\mathcal C$$. And a $$\mathcal D$$-morphism from $$i_1\colon A_1\to B$$ to $$i_2\colon A_2\to B$$ is a $$\mathcal C$$-morphism $$g\colon A_1\to A_2$$ such that $$i_2\circ g=i_1$$. Now the kernel is a $$\mathcal D$$-object such that there is exactly one $$\mathcal D$$-morphism to any $$\mathcal D$$-object. i.e., the kernel is an initial object in $$\mathcal D$$.
For other universal properties, we consider categories with more complex objects, but in general the objects are diagrams over the original category (here: just one arrow) with certain "boundary conditions" (here: codomain $$B$$ and $$f\circ i=0$$), and the morphisms are morphisms between such diagrams (i.e., collections of morphisms between corresponding points of two diagrams that make the created meshes commute), again perhaps with certain "boundary conditions" (here: we postulate that the identity is used at $$B$$, which we did implicitely above by not having an arrow $$B\to B$$ in the definition of $$\mathcal D$$-morphism to begin with). Then in this category we look for an initial (or terminal) object.