nLab categorical semantics

Idea

One may interpret mathematical logic as being a formal language for talking about the collection of monomorphisms into a given object of a given category: the poset of subobjects of that object.

More generally, one may interpret type theory and notably dependent type theory as being a formal language for talking about slice categories, consisting of all morphisms into a given object.

Conversely, starting with a given theory of logic or a given type theory, we say that it has a categorical semantics if there is a category such that the given theory is that of its slice categories, if it is the internal logic of that category.

Definition

For the general idea, for the moment see at type theory the section An introduction for category theorists and see at relation between type theory and category theory.

Of dependent type theory

We discuss how to interpret judgements of dependent type theory in a given category $\mathcal{C}$ with finite limits. For more see categorical model of dependent types.

Write $cod : \mathcal{C}^I \to \mathcal{C}$ for its codomain fibration, and write

$$\chi : \mathcal{C}^{op} \to Cat$$

for the corresponding classifying functor, the self-indexing

$$\chi : \Gamma \mapsto \mathcal{C}_{/\Gamma}$$

that sends an object of $\mathcal{C}$ to the slice category over it, and sends a morphism $f : \Gamma \to \Gamma'$ to the pullback/base change functor

$$f^\ast : \mathcal{C}_{/\Gamma'} \to \mathcal{C}_{/\Gamma} \,.$$

We give now rules for choices “$[x y z]$” that associate with every string “$x y z$” of symbols in type theory objects and morphisms in $\mathcal{C}$. A collection of such choices following these rules is an interpretation / a choice of categorical semantics of the type theory in the category $\mathcal{C}$.

Contexts and type judgements

1. The empty context $()$ in type theory is interpreted as the terminal object of $\mathcal{C}$

   $$[ () ] := * \,.$$

2. If $\Gamma$ is a context which has already been given an interpretation $[\Gamma] \in Obj(\mathcal{C})$, then a judgement of the form

   $$\Gamma \vdash A : Type$$

   is interpreted as an object in the slice over $[\Gamma]$

   $$[\Gamma \vdash A : Type] \in Obj(\mathcal{C}_{/\Gamma}) \,,$$

   hence as a choice of morphism

   $$\array{ [(\Gamma, x : A)] \\ \downarrow^{\mathrlap{[\Gamma \vdash A : Type]}} \\ [\Gamma] }$$

   in $\mathcal{C}$.

3. If a judgement of the form $\Gamma \vdash A : Type$ has already found an interpretation, as above, then an extended context of the form $(\Gamma, x : A)$ is interpreted as the domain object $[(\Gamma, x : A)]$ of the above choice of morphism.

Terms

Assume for a context $\Gamma$ and a judgement $\Gamma \vdash A : Type$ we have already chosen an interpretation $[\Gamma, x : A] \stackrel{[\Gamma \vdash A : Type]}{\to} [\Gamma]$ as above.

A judgement of the form $\Gamma \vdash a : A$ (a term of type $A$) is to be interpreted as a section of this morphism, equivalently as a morphism in $\mathcal{C}_{/[\Gamma]}$

$$[\Gamma \vdash a : A] : * \to [\Gamma, x : A]$$

from the terminal object to $[\Gamma \vdash A : Type]$, which in $\mathcal{C}$ is a commuting triangle

$$\array{ [\Gamma] &&\stackrel{[(\Gamma \vdash a : A)]}{\to}&& [\Gamma, x : A] \\ & {}_\mathllap{\mathrm{id}_{[\Gamma]}}\searrow && \swarrow_{\mathrlap{[\Gamma \vdash A : Type]}} \\ && [\Gamma] } \,.$$

Variables

For a term $\Gamma \vdash a : A$ the context $\Gamma$ is the collection of free variables in $a$.

(…)

Substitution

Assume that interpretations for judgements

$$\Gamma , x : A \vdash B(x) : Type$$

and

$$\Gamma \vdash a : A$$

have been given as above. Then the substitution judgement

$$\Gamma \vdash B[a/x] : Type$$

is to be interpreted as follows. The interpretation of the first two terms corresponds to a diagram in $\mathcal{C}$ of the form

$$\array{ &&&& [(\Gamma, x : A, y : B(x))] \\ &&&& \downarrow^{\mathrlap{[\Gamma, x : A \vdash B(x) : Type]}} \\ [\Gamma] &&\stackrel{[\Gamma \vdash a : A]}{\to}&& [(\Gamma, x : A)] \\ & {}_{id}\searrow && \swarrow_{\mathrlap{[\Gamma \vdash A : Type]}} \\ && [\Gamma] }$$

The interpretation of the substitution statement is then the pullback

$$[\Gamma \vdash B[a/x] : Type] := [\Gamma \vdash a : A]^* [\Gamma, x : A \vdash B(x) : Type] \,,$$

hence the morphism in $\mathcal{C}$ that universally completes the above diagram as

$$\array{ [(\Gamma, y : B[x/a])] &&\to&& [(\Gamma, x : A, y : B(x))] \\ {}^{\mathllap{[\Gamma \vdash B[a/x] : Type] }}\downarrow &&&& \downarrow^{\mathrlap{[\Gamma, x : A \vdash B(x) : Type]}} \\ [\Gamma] &&\stackrel{[\Gamma \vdash a : A]}{\to}&& [(\Gamma, x : A)] \\ & {}_{id}\searrow && \swarrow_{\mathrlap{[\Gamma \vdash A : Type]}} \\ && [\Gamma] }$$

Of homotopy type theory

See categorical semantics of homotopy type theory.

Examples

• models in presheaf toposes

Terminology

Most usage in mathematics of the adjective “categorical” in relation to category theory is a shorthand, and arguably an unfortunate one, for “category theoretic”, i.e. for “as seen through the lens of, hence as treated with the concepts and tools of category theory”.

Compare to “numerical” versus “number theoretic”: The interest of number theory is not in numerical methods! Numerical statements are made by engineers. For category theorists it’s worse: Their profession is not to do categorical arguments, not in the dictionary sense of “categorical” as “unambiguous, absolute, unqualified”.

While the careful dictionaries list a secondary sense of “categorical”, as “relating to a category”, they hardly have the categories of Kant in mind here, much less those of Eilenberg & MacLane; instead they refer to categorization: Merriam-Webster offers “categorical systems for classifying books” as an example for what “categorical” could refer to, in this secondary sense, in common language. Therefore it does not seem to help much that this secondary common sense of “categorical” is offered as the primary sense of “categorial” (without the second “c”!). But this is probably the rationale behind a more widespread use of “categorial semantics” over “categorical semantics” in the field of formal logic.

All this notwithstanding, the use in mathematics of “categorical”, as in categorical semantics is wide-spread, even standard. Since unfortunate choice of terminology in mathematics is rather common (compare “perverse schobers”!), and since the primary purpose of mathematical terminology is communication rather than, yes, proper categorization, there may not be much gain in opposing this trend, and not much success to doing so by replacing unfortunate terminology with something that looks like the same unfortunate terminology but with a typo in it.

Related pages

• relation between type theory and category theory

• categorical model of dependent types

• initiality conjecture

• Awodey's conjecture

References

A standard textbook reference for categorical semantics of logic is section D1.2 of

• Peter Johnstone, Sketches of an Elephant

The categorical semantics of dependent type theory in locally cartesian closed categories is essentially due to

• R. A. G. Seely, Locally cartesian closed categories and type theory, Math. Proc. Camb. Phil. Soc. (1984) 95 (pdf)

For more references on this see at relation between category theory and type theory.

Lecture notes on this include for instance.

• Martin Hofmann, Syntax and semantics of dependent types, Semantics and Logics of Computation (P. Dybjer and A. M. Pitts, eds.), Publications of the Newton Institute, Cambridge University Press, Cambridge, (1997) pp. 79-130. (web, )

• Roy Crole, Categories for types

See also section B.3 of

• Michael Warren, Homotopy Theoretic Aspects of Constructive Type Theory (pdf)

A comprehensive definition of semantics of homotopy type theory in type-theoretic model categories is in section 2 of

• Michael Shulman, Univalence for inverse diagrams and homotopy canonicity, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( From type theory and homotopy theory to Univalent Foundations of Mathematics ) June 2015 (arXiv:1203.3253, doi:/10.1017/S0960129514000565)