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I have been trying to formally state the axioms of category in predicate logic. It seems that I will need one-place predicates for objects and arrows, two-place predicates for heads and tails of arrows and a three-place predicate for composition. I am trying to avoid the use of the functional notation found in most definitions of CT, notation which IIUC more properly belongs in the domain of set theory.

EDIT: Predicate logic alone would seem to be completely unworkable as a language to express the axioms of category theory. I don't see any way around starting with 11 universally quantified variables in predicate logic for the axiom of associativity, compared to only 3 when using the '$\circ$' notation for the composition of arrows. (Sigh)

Dan Christensen
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  • Can you give an example of the functional notation to which you refer? – Cameron Buie Dec 29 '15 at 21:13
  • Functions or $k$-ary operations can be replaced by $k+1$-ary predicates, with suitable axioms of unique existence to express functional dependence of the "output" argument on the inputs. – hardmath Dec 30 '15 at 00:17
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    Note that extending first-order logic to allow partial functions is in some sense a "trivial" extension - you're just adding symbols to stand for first-order expressions. So you can use a partial function symbol for composition, and still retain all the niceness of first-order model theory (e.g. compactness, Lowenheim-Skolem, etc.) as long as you're careful. – Noah Schweber Dec 30 '15 at 05:00

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You can actually make do with a smaller language: one ternary relation (composition $Comp$), and two unary functions (source $s$ and target $t$). We conflate objects with identity arrows, so we don't need a separate sort for objects at all.

The axioms now are straightforward:

  • "Compositions behave correctly." For every pair of arrows $\alpha,\beta$, there is at most one arrow $\gamma$ such that $Comp(\alpha,\beta,\gamma)$; and such a $\gamma$ exists iff $s(\beta)=t(\alpha)$.

  • "The source and target of an identity arrow are itself." $t(t(\alpha))=s(t(\alpha))\wedge t(s(\alpha))=s(s(\alpha))$.

  • "Compositions are associative." This is a bit long. For instance, to write $(\alpha\beta)\delta=\alpha(\beta\delta)$, we'd write $Comp(\alpha, \beta,\gamma)\wedge Comp(\gamma, \delta, \epsilon)\wedge Comp(\beta, \delta,\theta)\implies Comp(\alpha, \theta, \epsilon).$ But it's totally doable.

  • "Identities are identities." $Comp(\alpha, \beta,\gamma)\wedge s(\alpha)=\alpha\implies \beta=\gamma$, and similarly for composition on the right.

  • "Identities are their own source and target." $t(\alpha)=\alpha\iff s(\alpha)=\alpha$.

And that's it!

This is not due to me; I read about it in Goldblatt's book Topoi. I forget who he attributed it to.

Noah Schweber
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  • Do we need a unary predicate for "identity arrows"? – hardmath Dec 29 '15 at 21:25
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    @hardmath No, an identity arrow is one satisfying $\alpha=s(\alpha)$ (or $\alpha=t(\alpha)$). – Noah Schweber Dec 29 '15 at 21:28
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    A variant on this definition (which even avoids including $s$ and $t$ in the language) is used in Freyd's book Abelian Categories, published in 1964. That might be the first place it was written down (it is for a surprising number of ideas in category theory!). – Eric Wofsey Dec 30 '15 at 05:06
  • Very helpful. Who ever knew that category theory was so simple! – Wildcard Aug 15 '16 at 06:34