Terminology and notation

Here is a brief statement of some of the terminology and notation we will be using in this post.<\p>

The terminology used in category theory may seem a little unusual.
In the table below are some of the terms used in category theory for certain types of arrow,
shown above the horizontal line in each cell of the table;
below that line is the term normally used
when speaking of functions in the category of sets.

Categorical terminology for arrows		Invertible? ($\longleftrightarrow$)
		Not necessarily	Yes
Loop? $\bigl(\circlearrowright\bigr)$	Not necessarily	arrow, morphism, map function	isomorphism bijection
	Yes	endomorphism endofunction, loop, self-map	automorphism permutation

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Notes:

1. “Permutation” sometimes connotes that the underlying set is finite.
2. “Bijection” is a synonym for the older term “one-to-one correspondence.”
3. “Arrow” is the most abstract of the terms.

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Terminology

Graph

We use the definition fairly standard in category theory,
that “graph” means directed graph,
which is simply a parallel pair of arrows in a category: $\mathop\rightrightarrows\limits^{d_0}_{d_1}$.
The object which is the source of the arrows is considered the object of edges;
the object which is the target of the arrows is considered the object of vertices.
The arrows $d_0,d_1$ determine respectively the source and target of the edges.
If the context makes it clear that the ambient category is the category of sets, $\Set$,
then this resembles the graph theorists definition of directed graph.
But see the nlab article for the many meanings of the term.

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Notation

Order for arrow composition

In most (I think all, but a slip-up is possible) instances in this blog,
when writing the composition of arrows in text,
I write the composition in diagrammatic, left-to-right order.
Thus the composite of $X \xrightarrow f Y$ with $Y \xrightarrow g Z$ would be written $X \xrightarrow {fg} Z$.

Links to detailed bibliographies for

Max Kelly

https://en.wikipedia.org/w/index.php?title=Max_Kelly&oldid=975556001

Ross Street

https://en.wikipedia.org/w/index.php?title=Ross_Street&oldid=975565027

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Some highlights (in my opinion) of this blog:
Basic examples of Kan extension in logic and set theory
Relations between limits in categories
The bimodule of sets, functions and permutations
Double groupoids and the twelvefold way
The identities-are-free (Yoneda) lemma
Yoneda structures 1
The category structures on 2
Relations between the categories Set and 2