nLab simplicial type theory

Idea

Simplicial type theory (Riehl & Shulman 2017, §3) is a homotopy type theory which introduces additional non-fibrant layers on top of Martin-Löf type theory to provide a logical calculus for a directed interval type and with it for geometric shapes, consisting of Segal space-types, Rezk complete-types, and covariant fibrations.

Formal presentations

There are many different ways to formalize the directed interval primitive in simplicial type theory:

• one could simply postulate in vanilla dependent type theory via inference rules and axioms a directed interval primitive as a bounded total order, as in Gratzer, Weinberger, & Buchholtz 2024 and Gratzer, Weinberger, & Buchholtz 2025.

• one could use two-level type theory consisting of a non-fibrant layer, and then axiomatise a directed interval primitive as a bounded total order in the non-fibrant layer

• one could use type theory with shapes consisting of a non-fibrant layer of cubes which is a simple type theory and a non-fibrant layer of topes which is a propositional logic over the cube layer, and then define the directed interval primitive as a bounded total order using the cube and tope layers, as in Riehl & Shulman 2017

With axioms

The directed interval primitive $\mathbb{2}$ in simplicial type theory is a non-trivial bounded total order. Bounded total orders have many definitions in mathematics, such as a bounded partial order satisfying totality, or a lattice satisfying totality. As such, there are multiple possible sets of inference rules one could use to present the directed interval in the formalization.

The axiomx for the directed interval primitive $(\mathbb{2}, \leq)$ as a partial order in the formalization are as follows:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{2} \; \mathrm{type}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{2}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 1:\mathbb{2}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash x \leq y \; \mathrm{type}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash \tau_\leq(x, y):\mathrm{isProp}(x \leq y)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{refl}_\leq(x):x \leq x} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2}, z:\mathbb{2}, p:x \leq y, q:y \leq z \vdash \mathrm{trans}_\leq(x, y, z, p, q):(x \leq z)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{antisym}_\leq(x, y):\mathrm{isEquiv}(idtoleqgeq(x, y))} \quad \mathrm{where} \quad \begin{array}{l} idtoleqgeq(x, y):(x =_\mathbb{2} y) \to (x \leq y) \times (y \leq x) \\ idtoleqgeq(x, x)(\mathrm{id}_\mathbb{2}(x)) \coloneqq (\mathrm{refl}_\leq(x), \mathrm{refl}_\leq(x)) \end{array}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{totl}_\leq(x, y):[(x \leq y) + (y \leq x)]} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{bot}_\leq(x):0 \leq x} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{top}_\leq(x):x \leq 1}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{nontriv}_\leq:(0 =_\mathbb{2} 1) \to \mathbb{0}}$$

The axioms for the directed interval primitive $\mathbb{2}$ as a lattice in the formalization are as follows:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{2} \; \mathrm{type}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \tau_0:\mathrm{isSet}(\mathbb{2})} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{2}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 1:\mathbb{2}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash x \wedge y:\mathbb{2}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash x \vee y:\mathbb{2}}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{idem}_\wedge(x):x \wedge x =_{\mathbb{2}} x} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{comm}_\wedge(x, y):x \wedge y =_{\mathbb{2}} y \wedge x}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2}, z:\mathbb{2} \vdash \mathrm{assoc}_\wedge(x, y, z):(x \wedge y) \wedge z =_{\mathbb{2}} x \wedge (y \wedge z)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{lunit}_\wedge(x):1 \wedge x =_{\mathbb{2}} x} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{runit}_\wedge(x):x \wedge 1 =_{\mathbb{2}} x}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{idem}_\vee(x):x \vee x =_{\mathbb{2}} x} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{comm}_\vee(x, y):x \vee y =_{\mathbb{2}} y \vee x}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2}, z:\mathbb{2} \vdash \mathrm{assoc}_\vee(x, y, z):(x \vee y) \vee z =_{\mathbb{2}} x \vee (y \vee z)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{lunit}_\vee(x):0 \vee x =_{\mathbb{2}} x} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2} \vdash \mathrm{runit}_\vee(x):x \vee 0 =_{\mathbb{2}} x}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{asorp}_\wedge(x, y):x \wedge (x \vee y) =_{\mathbb{2}} x} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{asorp}_\vee(x, y):x \vee (x \wedge y) =_{\mathbb{2}} x}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x:\mathbb{2}, y:\mathbb{2} \vdash \mathrm{totl}(x, y):[(x \wedge y =_{\mathbb{2}} x) + (x \wedge y =_{\mathbb{2}} y)]} \quad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{nontriv}_\leq:(0 =_\mathbb{2} 1) \to \mathbb{0}}$$

We inductively define the $n$-simplices as the following family of types:

• The $0$-simplex is the unit type

  $$\Delta^0 \coloneqq \mathbf{1}$$

• Let $\mathrm{Fin}(n) \coloneqq \sum_{i:\mathbb{N}} i \lt n$ be the canonical finite type with $n$ elements. The $n+1$-simplex is defined as the type

  $$n:\mathbb{N} \vdash \Delta^{n + 1} \coloneqq \sum_{t:\mathrm{Fin}(n + 1) \to \mathbb{2}} \prod_{i:\mathrm{Fin}(n)} t(\pi_1(i + 1)) \leq t(\pi_1(i))$$

The first few simplicies can be explicitly written as

$$\Delta^1 \coloneqq \mathbb{2}$$

$$\Delta^2 \coloneqq \sum_{t:\mathrm{Fin}(2) \to \mathbb{2}} t(1) \leq t(0)$$

$$\Delta^3 \coloneqq \sum_{t:\mathrm{Fin}(3) \to \mathbb{2}} (t(2) \leq t(1)) \times (t(1) \leq t(0))$$

Subshapes of the simplicies could also be defined.

• The boundary of the $1$-simplex is defined as

  $$\partial \Delta^1 \coloneqq \sum_{t:\Delta^1} [(t =_{\Delta^1} 0) + (t =_{\mathbb{2}} 1)]$$

• The boundary of the $2$-simplex is defined as

  $$\partial \Delta^2 \coloneqq \sum_{t:\mathrm{Fin}(2) \to \mathbb{2}} [(t(0) =_{\mathbb{2}} 0) \times (t(0) \leq t(1)) + (t(0) =_{\mathbb{2}} t(1)) + (t(0) \leq t(1)) \times (t(1) =_{\mathbb{2}} 1)]$$

Modalities

Simplicial type theory has recently been augmented with various modalities in the sense of modal type theory (Gratzer, Weinberger, & Buchholtz 2024, Gratzer, Weinberger, & Buchholtz 2025). These modalities include

• The op modality $A^\op$

• The global sections/core flat modality $\flat A$

• The sharp modality $\sharp A$

• The twisted arrow modality $A^\mathrm{tw}$

See also

• Reedy model structure

• bisimplicial set

• synthetic $(\infty,1)$-category theory

• dependent type theory

  • homotopy type theory

  • parametric type theory

  • modal type theory

  • type theory with shapes

  • two-level type theory

References

• Emily Riehl, Mike Shulman, A type theory for synthetic ∞-categories, Higher Structures 1 1 (2017) [arxiv:1705.07442, published article]

• Jonathan Weinberger, A Synthetic Perspective on $(\infty,1)$-Category Theory: Fibrational and Semantic Aspects $[$arXiv:2202.13132$]$

• Jonathan Weinberger, Strict stability of extension types $[$arXiv:2203.07194$]$

• Jonathan Weinberger, Generalized Chevalley criteria in simplicial homotopy type theory, arXiv:2403.08190

• Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, Directed univalence in simplicial homotopy type theory (arXiv:2407.09146)

• Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, The Yoneda embedding in simplicial type theory (arXiv:2501.13229)

A talk on synthetic (infinity,1)-category theory in simplicial type theory and infinity-cosmos theory:

• Emily Riehl, The synthetic theory of infinity-categories vs the synthetic theory of infinity-categories, Homotopy Type Theory Electronic Seminar Talks, 1 March 2018, (video, slides)

A proof assistant implementing simplicial type theory:

• Rzk[github.com/fizruk/rzk]

Formalization of the $(\infty,1)$-Yoneda lemma via simplicial homotopy type theory (in Rzk):

• Nikolai Kudasov, Emily Riehl, Jonathan Weinberger. Formalizing the $\infty$-categorical Yoneda lemma (2023) [arXiv:2309.08340]