1. Introduction
Defining the notion of
$\infty$
-category is a difficult task. There have been many different proposed definitions, to mention a few, by Boardman and Vogt (Boardman and Vogt Reference Boardman and Vogt1973) with the idea of quasi-categories, later developed by Joyal (Joyal Reference Joyal2008a) and Lurie (Lurie, Reference Lurie2009), or by Rezk in (Rezk Reference Rezk2001) in the form of complete Segal spaces. In the case of
$(\infty ,1)$
-categories, most of the known definitions have been shown to be equivalent in an appropriate homotopy theoretic sense; see, for example, the work by Toën (Toën Reference Toën2005), Bergner (Bergner Reference Bergner2007), and Joyal-Tierney (Joyal and Tierney Reference Joyal and Tierney2007).
Recent work by Riehl and Shulman (Riehl and Shulman Reference Riehl and Shulman2017) develops a synthetic theory of
$(\infty ,1)$
-categories. They achieve this by implementing an extension of homotopy type theory. Moreover, Riehl explores in (Riehl Reference Riehl2023) the novelty and advantages that support the philosophy behind this approach. We highlight some of these points here. First, all constructions are by design invariant under equivalence. Another aspect of this synthetic theory is that it is by construction compatible with the Univalence Axiom. Moreover, the standard model of the theory, the category of bisimplicial sets, has a well-known homotopy theory. Furthermore, it is possible to interpret this synthetic theory internally to any Grothendieck
$\infty$
-topos (see Remark 6.5). Finally, another pleasant consequence of this synthetic account is that it simplifies some definitions and constructions, streamlining the development of the theory of
$(\infty ,1)$
-categories. To begin with, the synthetic definition of an
$(\infty ,1)$
-category is fairly succinct in comparison with any other given before.
Previously, Voevodsky’s simplicial model for HoTT (Kapulkin and Lumsdaine Reference Kapulkin and Lumsdaine2021) interpreted types as
$\infty$
-groupoids. However, we do not have a similar interpretation of types as
$\infty$
-categories, where by
$\infty$
-category, we mean
$(\infty ,1)$
-category. To overcome this difficulty, (Riehl and Shulman Reference Riehl and Shulman2017) introduced a new type theory by adding to HoTT a strict interval object and the novel idea, due to Shulman and Lumsdaine (unpublished), of a new type former called extension type. We call this simplicial homotopy type theory, or sHoTT for short. This type theory can be seen as an instance of a cubical type theory; however, it is different from the one presented in (Cohen et al. Reference Cohen, Coquand, Huber and Mörtberg2015).
In this setting, it is possible to define Rezk types as certain special types that play the role of
$ \infty$
-categories. In this synthetic treatment, they retrieve that some expected results from higher categories, including a theory for adjunctions, the analog of left (right) fibrations, which are called covariant (contravariant) families, and the Yoneda lemma for these fibrations. Additional work by Buchholtz and Weinberger (Buchholtz and Weinberger Reference Buchholtz and Weinberger2023) studies synthetic cocartesian fibrations, a generalization of covariant families. Further treatment can be found in Weinberger’s PhD thesis (Weinberger Reference Weinberger2021). The cited work also includes two-sided fibrations.
Exploiting results from (Shulman Reference Shulman2015), it is shown in (Riehl and Shulman Reference Riehl and Shulman2017) that sHoTT has a model in bisimplicial sets where Rezk types correspond to complete Segal spaces (also called Rezk spaces).
The original presentation of sHoTT does not implement constructions such as the opposite of an
$ (\infty ,1)$
-category, the core
$\infty$
-groupoid of an
$ (\infty ,1)$
-category, and we lack of a Yoneda embedding in which an
$ (\infty ,1)$
-category is embedded into an
$ (\infty ,1)$
-category of presheaves of spaces. Another important problem is that the type of discrete types is a Segal type that does not coincide with the
$ \infty$
-category of
$ \infty$
-groupoids in the bisimplicial sets model. However, the last problem was solved working in (Weaver and Licata Reference Weaver and Licata2020) using bicubical sets. An alternative solution, along with the implementation of other constructions, is contained in the work of Gratzer–Buchholtz–Weinberger (Gratzer et al. Reference Gratzer, Weinberger and Buchholtz2024, Reference Gratzer, Weinberger and Buchholtz2025) by enhancing sHoTT with modalities (see Remark 5.15).
Despite the current inherent limitations of the type theory, we have tried to explore which other categorical properties can be obtained with the theory as is. Throughout this paper, we intend to present a reasonable theory of limits and colimits of diagrams of
$(\infty ,1)$
-categories. In this setting, we can prove most of the expected properties that are familiar from category theory. Under the mild assumption of the existence of a type-theoretic universe of spaces, we can compute the limit of a space-valued diagram as a dependent product (see Remark 5.15).
1.1 Limits and colimits
Limits and colimits have been studied extensively for quasi-categories in (Lurie Reference Lurie2009), and for Segal spaces. A short presentation appears in (Rasekh Reference Rasekh2023). We introduce the definitions of limits and colimits within this synthetic theory and verify that they are consistent with the current definition for Rezk spaces in the bisimplicial model of sHoTT. We also prove some expected results, such as Proposition 3.7, which can be phrased as the universal property of colimits. We also present the interaction between limits and colimits with adjoints, Theorem 3.9 showing that right adjoints preserve limits.
We prove in Corollary 4.6 that in a Rezk type, the type of all limits of a given diagram is a proposition. This can be understood as a uniqueness property. Last, in Section 5, we show that in an appropriate sense, any limit of spaces can be computed simply as a dependent product. The goal is to replicate the fact that for any diagram of spaces
$ G: I \to \infty \text{-}\mathbf{Gpd}$
, where
$ I$
is a set,
$ \lim _I G = \prod _{i\in I}G_i$
. The difficulty in carrying out this computation in sHoTT is that we are unable to construct the correct type of spaces (discrete types) within the theory. We implement this using a directed formulation of the Univalence Axiom, due to Cavallo, Riehl, and Sattler, which allows us to assume the existence of a type with the desired properties.
The reader will appreciate that the study of limits and colimits in the synthetic setting is relatively simple. Our only prerequisite is some knowledge of homotopy type theory. This is in line with the goal of having a synthetic theory of
$\infty$
-categories where these are somewhat basic objects out of which we can effortlessly develop a robust synthetic theory.
1.2 Outline
In Section 2, we begin with an introduction to the work of Riehl-Shulman. The material we present here is not exhaustive, but for the understanding and development of this work it will be enough. All results contained in this section are due to Riehl-Shulman and can be found in (Riehl and Shulman Reference Riehl and Shulman2017). The reader who is interested in the details of simplicial homotopy type theory is invited to read the mentioned reference; the experienced reader may skip this whole section.
Having the basic theory at hand, we define in Section 3 limits and colimits. We prove Proposition 3.7, which can be understood as the universal property of limits and colimits. We also prove the analogous result from category theory that right adjoints preserve limits and left adjoints preserve limits.
In Section 4, we study limits in Rezk types. This special case yields Corollary 4.6, which is the uniqueness of limits up to equality. Up to this point, all the results and proofs can be dualized by hand. Furthermore, in the presence of the opposite categories, this dualization can be justified formally c.f. (Gratzer et al. Reference Gratzer, Weinberger and Buchholtz2024). Finally, in Section 5, we carry out the computation of the limit of a diagram of “spaces” as a dependent product. Here, we use univalent covariant families, due to Cavallo, Riehl, and Sattler, to make sense of the
$\infty$
-category of spaces. Because simply taking the type of all Rezk types does not yield the correct object.
Finally, in Section 6, we verify that our definitions are consistent with the semantics. The general procedure we follow is to first interpret our type-theoretic definition in the intended semantics of bisimplicial sets and then prove that the resulting statement is equivalent to the existing definition. For limits and colimits, we do this in Subsection 6.1.
Computer-verified proofs.
While preparing this paper, a formalization project for sHoTT (Kudasov et al. Reference Kudasov, Riehl and Weinberger2023) came to light. The overall goal is to, within the present framework, obtain computer-verified proofs of results about
$ \infty$
-categories. Using the new proof assistant Rzk, the project aims to formalize the content of (Riehl and Shulman Reference Riehl and Shulman2017; Buchholtz and Weinberger Reference Buchholtz and Weinberger2023). The stub 13-limits.rzk.md of this project aims to formalize the content of the present work.
2. Extension Types, Segal Types, and Covariant Families
In this section, we introduce the necessary material of the type theory that we use in the paper. We refrain from providing detailed proofs. The best reference for this is the original paper (Riehl and Shulman Reference Riehl and Shulman2017) where simplicial homotopy type theory was first introduced. We indicate precisely where each result can be found in ibid.
2.1 Extension types
Essentially, simplicial homotopy type theory is ordinary homotopy type theory augmented with some axioms postulating the existence of a strict interval object
$ \unicode{x1D7DA}$
. That is, a type
$ \unicode{x1D7DA}$
equipped with a total order, a smallest element
$0$
and a distinct largest element
$1$
. In a general type
$X$
, an arrow in
$X$
can be defined as a map
$f: \unicode{x1D7DA} \to X$
, with the source and target of
$f$
being the image of the endpoints
$0, 1 : \unicode{x1D7DA}$
. However, we would like to be able to talk about the family of hom types from
$x$
to
$y$
, for
$x,y : X$
. This can be done in ordinary type theory at the expense of adding extra equalities
$f(0)=x$
and
$f(1)=y$
. The new type former, called extension type, forces definitional equalities at the endpoints. This type former makes it easier to handle arrows in the type
$X$
. However, the inner workings of extension types force us to single out the interval
$ \unicode{x1D7DA}$
by putting it in a separate layer of a layered type theory.
More concretely, simplicial type theory is built as a three-layer type theory: the layer of cubes, the layer of topes, and the layer of types. All rules for each layer can be found in greater detail in our main reference (Riehl and Shulman Reference Riehl and Shulman2017), here, we just give a small account together with some of the fundamental results we will need.
The first layer is a type theory with finite products of types. Some rules of this type theory include the following:
\begin{equation*} \frac {}{{\mathbf{1}} { cube}} \quad \quad \frac {}{\Gamma \vdash \star : {\textbf {1}}} \end{equation*}
\begin{equation*} \frac {I \, {cube} \quad J \, {cube}}{I\times J { cube}.} \end{equation*}
The second layer is an intuitionistic logic over the layer of cubes. Its types are called topes. Topes can be regarded as polytopes embedded in a cube. They admit operations of finite conjunction and disjunction, but negation, implication, or quantifiers are a not part of this theory. The complete rules for topes can be found in Riehl and Shulman (Reference Riehl and Shulman2017). There is an “equality tope” that appears in the third layer as a “strict equality.” For example, this tope equality is used to define some shapes below, and it is denoted with the symbol “
$\equiv$
.”
In simplicial type theory, the cubes layer is given as follows: It starts by postulating the cube
$ \unicode{x1D7DA}$
, which has two elements
$ 0: \unicode{x1D7DA}$
and
$ 1: \unicode{x1D7DA}$
. This cube also comes with an inequality tope
\begin{equation*} \frac {x: \unicode{x1D7DA} \quad y: \unicode{x1D7DA}}{(x\leq y)\,{\mathsf {tope}}}. \end{equation*}
In addition, there are axioms that turn the inequality tope into a total order relation on
$ \unicode{x1D7DA}$
with distinct endpoints
$0$
and
$1$
. Recall that these axioms are:
\begin{equation*} \frac {x: \unicode{x1D7DA}}{\vdash x\leq x}\quad \quad \frac {x: \unicode{x1D7DA},y: \unicode{x1D7DA},z: \unicode{x1D7DA}}{(x\leq y),(y\leq z) \vdash (x\leq z)}\quad \quad \frac {x: \unicode{x1D7DA},y: \unicode{x1D7DA}}{(x\leq y),(y\leq x) \vdash (x\equiv y)} \end{equation*}
\begin{equation*} \frac {x: \unicode{x1D7DA},y: \unicode{x1D7DA}}{\vdash (x\leq y)\vee (y\leq x)}\quad \quad \frac {x: \unicode{x1D7DA}}{\vdash (0\leq x)}\quad \quad \frac {x: \unicode{x1D7DA}}{\vdash (x\leq 1)}\quad \quad \frac {(0\equiv 1)}{\vdash \bot }. \end{equation*}
The rest of the cubes are generated by finite products of the cube
$ \unicode{x1D7DA}$
. Using cubes and topes, we can introduce shapes as follows:
\begin{equation*} \frac {I\quad {\mathsf { cube}} \quad t:I\vdash \phi \quad {\mathsf {tope}}}{\{t:I\: | \:\phi \}\quad {\mathsf {shape}}.} \end{equation*}
Some relevant shapes are:
$ \Delta ^0:\equiv \{t:{\textbf {1}}|\top \},$
$ \Delta ^1:\equiv \{t: \unicode{x1D7DA}|\top \},$
$ \Delta ^2:\equiv \{\langle t_1,t_2\rangle : \unicode{x1D7DA}\times \unicode{x1D7DA}|t_2\leq t_1 \},$
$ \partial \Delta ^1:\equiv \{t: \unicode{x1D7DA}|(t\equiv 0)\vee (t\equiv 1)\},$
$ \partial \Delta ^2:\equiv \{\langle t_1,t_2\rangle :\Delta ^2| (0\equiv t_2\leq t_1)\vee (t_1\equiv t_2)\vee ( t_2\leq t_1 \equiv 1)\},$
$ \Lambda _1^2:\equiv \{\langle t_1,t_2\rangle :\Delta ^2| (t_1\equiv 1)\vee (t_2\equiv 0)\}.$
More can be said about this strict interval: the category of simplicial sets is the classifying topos for such strict intervals (Maclane and Moerdijk Reference Maclane and Moerdijk1994). Furthermore, as explained in (Riehl and Shulman Reference Riehl and Shulman2017), by embedding simplicial sets into the category of bisimplicial sets, it is possible to show that the latter presents the “classifying (
$ \infty$
,1)-topos” of strict intervals.
Finally, there is a third layer of types that has all the ordinary type formers of homotopy type theory and one additional type former, the extension type, that involves the previous layers. Its formation rule is:
\begin{equation*} \frac {\begin{matrix} \{t:I \:| \:\phi \}\,{\mathsf {shape}}\\ \Xi \:| \:\Phi \vdash \Gamma \, {\mathsf {context}}\\ \end{matrix} \quad \begin{matrix} {\{t:I \:| \:\psi \}\,{\mathsf {shape}}}\\ {\Xi , t:I \:| \:\Phi ,\psi \:| \:\Gamma \vdash A\,{\mathsf {type}}}\\ \end{matrix} \quad \begin{matrix} {t:I \:| \:\phi \vdash \psi }\\ {\Xi , t:I \:| \:\Phi ,\phi \:| \:\Gamma \vdash a:A}\\ \end{matrix} }{\Xi \:| \: \Phi \: | \: \Gamma \vdash \left \langle \prod _{t:I \:| \:\psi }A(t)\big |_{a}^{\phi } \right \rangle \,{\mathsf {type}}. } \end{equation*}
We refer to (Riehl and Shulman Reference Riehl and Shulman2017) for the precise formulation of its rules. The name extension types are suggestive of how to construe them. We can think of
$\{t:I \:| \:\phi \}$
as a “sub-shape” of
$\{t:I\: | \:\psi \}$
and read the judgment
$\Xi , t:I \,\, | \,\, \Phi ,\phi \: | \:\Gamma \vdash a:A$
as a function
$\phi \rightarrow A$
. We could represent a point in an extension type with a dashed arrow in the commutative diagram:
This dashed arrow does not have to be unique in any sense. The benefit of this diagrammatic representation of extension types will be obvious later on when we introduce Segal types. Note that the
$\mathsf {hom}$
type in Definition 2.7 is a special case of an extension type. As noted in (Buchholtz and Weinberger Reference Buchholtz and Weinberger2023), it is possible to obtain a similar theory by considering HoTT plus a strict interval theory and replacing extension types using identity types (see [(Riehl and Shulman Reference Riehl and Shulman2017), Pag. 119] and [(Buchholtz and Weinberger Reference Buchholtz and Weinberger2023), Section 2.4]), but this makes the theory harder to use. In this paper, we will follow the original Riehl-Shulman formulation of sHoTT using extension types.
Remark 2.1. There is a connection of sHoTT with cubical type theory (Cohen et al. Reference Cohen, Coquand, Huber and Mörtberg2015). We refer to [(Riehl and Shulman Reference Riehl and Shulman2017), Remark 3.2] where this is explained further. We just mention that the cubical path type
$\mathsf {Path}_A(x,y)$
can be expressed as the extension type
$\langle \prod _{t:\mathbb{I}}A |_{ {rec}_\vee (x,y)}^{t \equiv 0 \vee t\equiv 1} \rangle$
. This is analogous to the
$\mathsf {hom}$
types, we introduce below in Definition 2.7.
Extension types behave very much like dependent product types. In particular, one can think of extension types as
$\prod$
-types with a judgmental equality added to them. Furthermore, many results that hold for
$\prod$
-types are also valid for extension types. For example:
Theorem 2.2.
If
$t:I\: | \:\phi \vdash \psi$
,
$X:\{t:I\: | \:\psi \}\rightarrow \mathcal{U}$
and
$Y:\prod \limits _{t:I\: | \:\psi }(X\rightarrow U)$
, while
$a:\prod _{t:I\: | \:\psi }X(t)$
and
$b:\prod _{t:I\: | \:\phi }Y(t,x(t))$
, then
\begin{align*} \left \langle \prod _{t:I\: | \:\psi }(\sum _{x:X(t)}Y(t,x))\big |_{\lambda t.(a(t),b(t))}^{\phi }\right \rangle \simeq \left (\sum _{f:\left \langle \prod _{t:I\: | \:\psi }X(t)\big |_a^{\phi }\right \rangle }\left \langle \prod _{t:I\: | \:\psi }Y(t,f(t))\big |_f^{\psi }\right \rangle \right ). \end{align*}
Proof. This is [(Riehl and Shulman Reference Riehl and Shulman2017), Theorem 4.3].
This resembles [(The Univalent Foundations Program 2013), Theorem 2.15.7] that states the commutativity of
$\sum$
and
$\prod$
types. The functions needed for the proof come from introduction and computation rules for extension types; these rules are quite similar to those of dependent products. We will need this idea of the proof later in Lemma 4.1. Also, we have
Theorem 2.3. ([(Riehl and Shulman Reference Riehl and Shulman2017), Theorem 4.4]) Suppose that we have
$t:I\: | \:\phi \vdash \psi$
,
$t:I\: | \:\psi \vdash \chi$
,
$X:\{t:I\: | \:\chi \}\to \mathcal{U}$
and
$a:\prod \limits _{t:I\: | \:\phi }X(t)$
. Then
\begin{equation*}\left \langle \prod \limits _{t:I\: | \:\chi }X\big |_{a}^{\phi }\right \rangle \simeq \sum _{f:\left \langle \prod _{t:I\: | \:\psi }X\big |_a^{\phi }\right \rangle }\left \langle \prod _{t:I\: | \:\chi }X\big |_f^{\psi }\right \rangle .\end{equation*}
2.1.1 Extensionality
In homotopy type theory, we use the function extensionality axiom for dependent functions. We recall this formulation.
Axiom 2.4.
Let
$B:A \to \mathcal{U}$
be a dependent type family over a type
$A$
. For
$f,g: \prod _{a:A}B(a)$
, the canonical map
$f=g \to \prod _{a:A}(fa=ga)$
is an equivalence.
Note that the equality on the left is in the dependent product. An element of
$\prod _{a:A}(fa=ga)$
is referred to as a homotopy and is written as
$f \sim g$
. There is a version of function extensionality for extension types.
Axiom 2.5.
Suppose
$t:I \; | \; \phi \vdash \psi$
and that
$A:\{t:I \; | \; \psi \} \to \mathcal{U}$
is such that each
$A(t)$
is contractible, and moreover, there is an element
$a: \prod _{t:I \; | \; \phi }A(t)$
, then
$\left \langle \prod _{t:I \; | \; \psi }A(t)|_a^\phi \right \rangle$
is contractible.
Further discussion and different formulations on this axiom can be found in [(Riehl and Shulman Reference Riehl and Shulman2017), 4.4]. Moreover, taking
$\phi$
to be
$\bot$
in Axiom 2.5 gives us Axiom 2.4. This is because for
$\phi \equiv \bot$
extension types behave like ordinary dependent products (see [(Riehl and Shulman Reference Riehl and Shulman2017), Section 2.2]). In the paper, we assume Axiom 2.5, hence, we also have function extensionality, and we use the notation
$f\sim g$
from above.
2.2 Segal types
With extension types and simplices at hand, it is feasible to introduce “morphisms” of a type as “points” in a “hom space.” The coherences that witness “composition” between these morphisms are elements in another suitable type. We can use the above to define Segal types.
Let us recall first that from the type elimination for tope disjunction rules presented in Riehl and Shulman (Reference Riehl and Shulman2017), it is possible to prove the following: Given a type
$A$
, in order to construct an element
$a:A$
in context
$\partial \Delta ^1$
, we can just give two elements
$x,y:A$
, and vice versa, any two elements
$x,y:A$
determine another element
$a:A$
in context
$\partial \Delta ^1$
. Similarly, the construction of an element
$a:A$
in context
$\partial \Delta ^2$
is equivalent to giving three terms
$a_0,a_1,a_2:A$
in context
$t: \unicode{x1D7DA}$
such that
$a_0[0/t]\equiv a_1[0/t],$
$a_0[1/t]\equiv a_2[0/t]$
, and
$a_1[1/t]\equiv a_2[1/t]$
. The last construction can be seen as the “boundary of a triangle in
$A$
,” see Definition 2.8 below.
Remark 2.6. The type elimination for tope disjunction rules [(Riehl and Shulman Reference Riehl and Shulman2017), Figure 3] governs how the three layers, cubes, topes, and the intentional part, of the type theory interact. This allows to, for example, make sense of the type of functions
$ \phi \to A$
for a shape
$\phi$
and a type
$A$
over some context of cubes. For more details on this, we refer to [(Riehl and Shulman Reference Riehl and Shulman2017), Section 3.2].
Definition 2.7.
Given
$x,y:A$
, determining a term
$[x,y]:A$
in context
$\partial \Delta ^1$
, define
\begin{align*} {\mathsf {hom}}_A(x,y):\equiv \left \langle \Delta ^1\rightarrow A\big |_{[x,y]}^{\partial \Delta ^1}\right \rangle . \end{align*}
A term in this type is called an
arrow
in
$A$
.
Note that this is just an extension type where the type family has constant value
$A$
. An element
$f:{\mathsf {hom}}_A(x,y)$
has the property that
$f(0)\equiv x$
and
$f(1)\equiv y$
. Thus, such an element captures the idea of what an arrow in
$A$
from
$x$
to
$y$
should be.
Definition 2.8.
For
$x,y,z:A$
and
$f:{\mathsf {hom}}_A(x,y),g:{\mathsf {hom}}_A(y,z)$
and
$h:{\mathsf {hom}}_A(x,z)$
, there is a term
$[x,y,z,f,g,h]:A$
in context
$\partial \Delta ^2$
, define
For an interpretation of the types above, we use the bisimplicial sets model of sHoTT. We give more details in Subsection 6.1.
Definition 2.9.
A
Segal type
is a type
$A$
such that for all
$x,y,z:A$
and
$f:{\mathsf {hom}}_A(x,y),g:{\mathsf {hom}}_A(y,z)$
the type
is contractible.
Informally,
$A$
is a Segal type if any pair of composable arrows has an essentially unique composite.
The first component of the center of contraction of this type is the composition of
$f$
and
$g$
. We adopt the usual notation
$g\circ f$
for such a term. This composition of arrows is associative whenever it is defined. For any type
$ A$
, we can define
${\mathsf {id}}_x:{\mathsf {hom}}_A(x,x)$
as
${\mathsf {id}}_x(t)\equiv x$
for all
$t:\Delta ^1$
. Furthermore, under the assumption that
$A$
is a Segal type, this element is the identity for the composition (see [(Riehl and Shulman Reference Riehl and Shulman2017), Section 5.2 and 5.3]). We can also get a useful characterization of Segal types:
Theorem 2.10. ([(Riehl and Shulman Reference Riehl and Shulman2017), Theorem 5.5]) A type
$A$
is a Segal type if and only if the restriction map
\begin{equation*} (\Delta ^2 \to A) \to (\Lambda _1^2 \to A) \end{equation*}
is an equivalence.
Moreover, this allows us to show that if
$A:X\rightarrow \mathcal{U}$
is a family over a type or shape
$X$
such that for all
$x:X$
the type
$A(x)$
is a Segal type then the dependent product of the family
$A$
over
$X$
is again a Segal type [(Riehl and Shulman Reference Riehl and Shulman2017), Corollary 5.6]. This means that the dependent product of Segal types is again a Segal type.
Let
$A,B$
be types and
$\phi :A\rightarrow B$
a function. For any
$x,y:A$
, there is an induced function
\begin{align*} \phi _{\#}:\hom _A(x,y)\rightarrow \hom _B(\phi (x),\phi (y)) \end{align*}
defined by precomposition. We use
$\phi$
for this induced function instead of
$\phi _{\#}$
when there is no risk of confusion. We have:
Proposition 2.11.
Any function
$\phi : A \rightarrow B$
between Segal types preserves identities and composition. This is for all
$a,b,c,d:A$
and
$f:{\mathsf {hom}}_A(a,b),\, g:{\mathsf {hom}}_A(b,c)$
, then we have
$\phi (g\circ f)=\phi (g)\circ \phi (f)$
and
$\phi ({\mathsf {id}}_d)={\mathsf {id}}_{\phi (d)}$
.
Proof. See [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 6.1].
This proposition makes the functorial behavior of functions between Segal types apparent. From Theorem 2.10, the type of functions
$A\rightarrow B$
is a Segal type when
$B$
is itself a Segal type and
$A$
is a shape or type. Given two functions,
$f,g:A\rightarrow B$
we can select a term
$\alpha :{\mathsf {hom}}_{A\rightarrow B}(f,g)$
. Spelling out the definition of such a term we get for all
$s: \unicode{x1D7DA}$
a function
$\alpha (s):A\rightarrow B$
such that
$\alpha (0)\equiv f$
and
$\alpha (1)\equiv g$
. In addition, for all
$a:A$
, we have
\begin{equation*} \lambda (s: \unicode{x1D7DA}).\alpha (s)(a):{\mathsf {hom}}_B(fa,ga). \end{equation*}
We denote this function as
$\alpha _a$
and we refer to it as the component of
$\alpha$
at
$a:A$
. An element
$\alpha :{\mathsf {hom}}_{A\rightarrow B}(f,g)$
is called a natural transformation between
$f$
and
$g$
. Such terminology is supported by the following remarks:
Remark 2.12. Let
$A$
be a type or shape,
$B$
a Segal type and
$f,g:A\rightarrow B$
. Then, the induced function
\begin{equation*} {\mathsf {hom}}_{A\rightarrow B}(f,g)\rightarrow \prod \limits _{a:A}{\mathsf {hom}}_B(fa,ga)\end{equation*}
is an equivalence [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 6.3]. And also [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 6.5], if
$\alpha :{\mathsf {hom}}_{A\rightarrow B}(f,g),\beta :{\mathsf {hom}}_{A\rightarrow B}(g,h)$
and
$x:A$
then
\begin{equation*}(\beta \circ \alpha )_x=\beta _x\circ \alpha _x\text{ and } ({\mathsf {id}}_f)_x={\mathsf {id}}_{f(x)}.\end{equation*}
This is to say that natural transformations are component-wise defined and their composition is performed point-wise.
Remark 2.13. Under the same assumptions of the previous remark, if
$\alpha :{\mathsf {hom}}_{A\rightarrow B}(f,g)$
and
$k:{\mathsf {hom}}_{A}(x,y)$
then
$\alpha _y\circ fk=gk\circ \alpha _x$
.
Hence, we can say that a natural transformation is truly natural in the categorical sense.
2.2.1 Discrete types
Definition 2.14. [(Riehl and Shulman Reference Riehl and Shulman2017), Definition 7.1] For any type
$ A$
, there is a map
\begin{equation*} \mathsf{idtoarr}:\prod _{x,y:A} (x=_{A}y) \to {\mathsf {hom}}_{A}(x,y) \end{equation*}
given through path induction as
$ \mathsf{idtoarr}_{x,x}({\mathsf {refl}}_x):\equiv {\mathsf {id}}_x$
. We say that the type
$ A$
is discrete if
$ \mathsf{idtoarr}_{x,y}$
an equivalence for all
$ x,y:A$
.
The first aspect to note about discrete types is that they are also Segal types [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 7.3]. Moreover, we think of discrete types as synthetic
$\infty$
-groupoids.
2.2.2 Rezk types
In the category of bisimplicial sets, Segal spaces are, intuitively, bisimplicial sets containing categorical and homotopical information. These data are not necessarily compatible. Complete Segal spaces fix this problem. More precisely, adopting the terminology and notation of (Rezk Reference Rezk2001), we have
$X_{\text{hoequiv}}$
as the space of homotopy equivalences of
$X$
. This space
$X_{\text{hoequiv}}$
is contained in
$X_1$
and consists of homotopy equivalences. Then, there is an obvious map
$u:X_0 \to X_{\text{hoequiv}}$
, if
$ u$
is an equivalence of spaces, we say that the Segal space is complete. The map
$u$
is not always a weak equivalence; an example of such a Segal space can be found in [(Rasekh Reference Rasekh2018), Example 2.57]. A similar phenomenon occurs for Segal types. These have an associative, unital composition but are not “univalent,” just as Segal spaces are not necessarily complete. To amend this problem and obtain “better behaved” types, we need to introduce Rezk types.
Let
$A$
be a Segal type and
$f:{\mathsf {hom}}_A(x,y)$
, we define the type
\begin{equation*}\mathsf {isiso}(f):\equiv \left ( \sum \limits _{g:{\mathsf {hom}}_A(y,x)}g\circ f={\mathsf {id}}_x \right ) \times \left ( \sum \limits _{h:{\mathsf {hom}}_A(y,x)}f\circ h={\mathsf {id}}_y \right )\end{equation*}
and say that
$f$
is an isomorphism if this type is inhabited. The reader might note that we do not ask for the left and right inverses of an isomorphism to be equal. The reason behind this can be found in [(Riehl and Shulman Reference Riehl and Shulman2017), Section 10.1]. The type
$\mathsf {isiso}(f)$
is a proposition [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 10.2]. Therefore, we can define the type of isomorphisms between any two terms
$x,y$
in a Segal type
$A$
as
\begin{equation*}x\cong y:\equiv \sum \limits _{f:{\mathsf {hom}}_A(x,y)}\mathsf {isiso}(f).\end{equation*}
There is an immediate characterization of isomorphisms:
Proposition 2.15.
Let
$A$
be a Segal type and
$f : {\mathsf {hom}}_A (x, y)$
. Then,
$f$
is an isomorphism if and only if we have
$g : {\mathsf {hom}}_A (y, x)$
with
$g \circ f = {\mathsf {id}}_x$
and
$f \circ g = {\mathsf {id}}_y$
.
Proof. This is [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 10.1].
Given a Segal type
$A$
and
$x:A$
is clear that
${\mathsf {id}}_x$
is an isomorphism. We get a family of functions
\begin{equation*}{\mathsf {idtoiso}}_A:\prod \limits _{x,y:A} \left ( x=y\rightarrow x\cong y \right )\end{equation*}
constructed by path induction via
${\mathsf {idtoiso}}_A({\mathsf {refl}}_x):\equiv {\mathsf {id}}_x$
. Although the function
${\mathsf {idtoiso}}_A$
is dependent on the type
$A$
, we will drop
$A$
from the notation since it is always clear over which type the function is defined.
Definition 2.16.
A Segal type
$A$
is a
Rezk type
if
$\mathsf {idtoiso}$
is an equivalence. We denote the inverse by
$ {\mathsf {rezk}}$
.
We will need the following fact about Rezk types from [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 10.9]
Proposition 2.17.
If
$ B$
is a Rezk type then for any type or shape
$ X$
, the type
$ B^X$
is a Rezk type.
2.3 Covariant families
Left fibrations were introduced by Joyal in (Joyal Reference Joyal2008b), and further studied in (Lurie Reference Lurie2009). These are the quasi-categorical versions of functors cofibered in groupoids. In turn, the category of bisimplicial sets also has a well-known theory of left fibrations (see (Rasekh Reference Rasekh2023)). In sHoTT, we study them as covariant families, and they play a central role in subsequent sections.
Let
$C:A\rightarrow \mathcal{U}$
be a type family over a type
$A$
. Two elements
$x,y:A$
,
$f:{\mathsf {hom}}_A(x,y)$
and
$u:C(x),v:C(y)$
define the type
\begin{equation*}{\mathsf {hom}}_{C(f)}(u,v):\equiv \left \langle \prod \limits _{t: \unicode{x1D7DA}}C(f(t))\big |_{[u,v]}^{\partial \Delta ^1}\right \rangle .\end{equation*}
This type represents the type of arrows from
$u$
to
$v$
over
$f$
in the total type of
$C$
.
Definition 2.18.
A type family
$C:A\rightarrow \mathcal{U}$
over a Segal type
$A$
is
covariant
if for any
$f:{\mathsf {hom}}_A(x,y)$
and
$u: C(x)$
the type
\begin{equation*} \sum \limits _{v:C(y)}{\mathsf {hom}}_{C(f)}(u,v) \end{equation*}
is contractible. Similarly, if for any
$f:{\mathsf {hom}}_A(x,y)$
and
$v: C(y)$
the type
\begin{equation*} \sum \limits _{u:C(x)}{\mathsf {hom}}_{C(f)}(u,v) \end{equation*}
is contractible, we say that
$C$
is
contravariant
.
In the situation of the definition above, we denote the center of contraction of each type as
$(f_*u,{\mathsf {trans}}_{f,u})$
and
$(f^*v,{\mathsf {trans}}_{f,v})$
, respectively.
From [(Riehl and Shulman Reference Riehl and Shulman2017), Remark 8.3], we have that if
$g: B \rightarrow A$
is a function and
$C : A \rightarrow \mathcal{U}$
is a covariant type family, then
$\lambda b. C(g(b)) : B \rightarrow \mathcal{U}$
is also covariant. This just means that a covariant family is stable under substitution.
The first important example of a covariant family is given by the
$\mathsf {hom}$
type (see [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 8.13]):
Proposition 2.19.
Let
$A$
be a type and fix
$a : A$
. Then, the type family
$\lambda x. {\mathsf {hom}}_A (a, x) : A \rightarrow \mathcal{U}$
is covariant if and only if
$A$
is a Segal type.
We can show that if
$C:A\rightarrow \mathcal{U}$
is a covariant family over a Segal type
$A$
, then, the total type
\begin{equation*} \sum \limits _{x:A}C(x) \end{equation*}
is a Segal type [(Riehl and Shulman Reference Riehl and Shulman2017), Theorem 8.8]. Also, as previously advertised, if the covariant family is defined over a Segal type then it is functorial in the following sense:
Proposition 2.20.
Suppose
$A$
is a Segal type and
$C : A \rightarrow \mathcal{U}$
is covariant. Then, given
$f : {\mathsf {hom}}_A (x, y)$
,
$g : {\mathsf {hom}}_A (y, z)$
, and
$u : C(x)$
, we have
$g_* (f_* u) = (g\circ f )_* u$
and
$({\mathsf {id}}_x )_*u = u$
.
Proof. This is [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 8.16].
Furthermore, naturality holds:
Proposition 2.21.
Let
$C, D : A \rightarrow \mathcal{U}$
be two covariant families and a fiber-wise map
$\phi :\prod \limits _{x:A}C(x)\rightarrow D(x)$
. Then, for any
$f:{\mathsf {hom}}_A(x,y)$
and
$u : C(x)$
,
\begin{equation*} \phi _y (f_* u) = f_* (\phi _x(u)). \end{equation*}
Proof. See [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 8.17].
If we have a covariant family
$C:A\rightarrow \mathcal{U}$
over a Segal type, a path
$p:x=y$
and
$u:C(x)$
, then, the element
$({\mathsf {idtoiso}}(p))_*u:C(y)$
can be understood as transporting
$u$
along the arrow
${\mathsf {idtoiso}}(p)$
. The covariant transport and the usual type-theoretic transport along paths coincide; by this we mean that they are propositionally equal:
Lemma 2.22.
If
$A$
is Segal and
$C : A\rightarrow \mathcal{U}$
is covariant and
$e : x =_A y$
, then for any
$u : C(x)$
, we have
\begin{align*} {\mathsf {idtoiso}}(e)_* u = {\mathsf {transport}}^C (e, u). \end{align*}
Proof. [(Riehl and Shulman Reference Riehl and Shulman2017), Lemma 10.7].
2.3.1 Yoneda lemma
The Yoneda lemma is a fundamental result of category theory. In the context of
$\infty$
-categories, it has been proved for different models, such as quasi-categories in (Lurie Reference Lurie2009), (Joyal Reference Joyal2008a), for Segal spaces in (Rasekh Reference Rasekh2023) and
$\infty$
-cosmoi in (Riehl and Verity Reference Riehl and Verity2017). In simplicial type theory, as noted in (Riehl and Shulman Reference Riehl and Shulman2017), the result can be viewed as a directed version of path transport. Also, there is a dependent version of the Yoneda lemma, which can be seen as the analog to path induction for covariant families. Although we will need only one of its consequences, namely, representability, we quickly review it in full generality.
Let
$C:A\rightarrow \mathcal{U}$
a covariant family and a fix term
$a:A$
. Then, there are functions
\begin{equation*}{\mathsf {evid}}_a^C:\equiv \lambda \phi .\phi (a,{\mathsf {id}}_a):\left (\prod \limits _{x:A}{\mathsf {hom}}_A(a,x)\rightarrow C(x)\right )\rightarrow C(a),\end{equation*}
\begin{equation*}{\mathsf {yon}}_a^C:\equiv \lambda u.\lambda x.\lambda f.f_*u: C(a)\rightarrow \left (\prod \limits _{x:A}{\mathsf {hom}}_A(a,x)\rightarrow C(x)\right ).\end{equation*}
Theorem 2.23. (Yoneda lemma, [(Riehl and Shulman Reference Riehl and Shulman2017), Theorem 9.1]) If
$A$
is a Segal type, then for any covariant family
$C:A\rightarrow \mathcal{U}$
and
$a:A$
, the functions
${\mathsf {yon}}_a^C$
and
${\mathsf {evid}}_a^C$
are mutual inverses.
We have the dependent version of Yoneda:
Theorem 2.24. ([(Riehl and Shulman Reference Riehl and Shulman2017), Theorem 9.5]) If
$A$
is a Segal type,
$a:A$
and
$C:\prod \limits _{x:A}({\mathsf {hom}}_A(a,x)\rightarrow \mathcal{U})$
is covariant, then the function
\begin{equation*}{\mathsf {evid}}_a^C:\equiv \lambda \phi .\phi (a,{\mathsf {id}}_a):\left (\prod \limits _{x:A}\prod _{f:{\mathsf {hom}}_A(a,x)}C(x,f)\right )\rightarrow C(a,{\mathsf {id}}_a)\end{equation*}
is an equivalence.
One particular instance of this equivalence is when
$C:\equiv {\mathsf {hom}}_A(a',-)$
, where
$A$
is a Segal type and
$a':A$
. This gives the so-called Yoneda embedding. The concept of representability is closely related to this embedding.
Definition 2.25.
A covariant family
$C:A\rightarrow \mathcal{U}$
over a Segal type is
representable
if there exists
$a:A$
and a family of equivalences
\begin{align*} \prod \limits _{x:A}({\mathsf {hom}}_A(a,x)\simeq C(x)). \end{align*}
We conclude this section with a result that is of particular interest to us:
Definition 2.26.
A term
$b:B$
is
initial
if for any
$x:B$
the type
${\mathsf {hom}}_B(b,x)$
is contractible. That is, if the type
\begin{align*} {\mathsf {isinitial}}(b):\equiv \prod \limits _{x:B}{\mathsf {iscontr}}({\mathsf {hom}}_B(b,x)) \end{align*}
is inhabited. Also, a term
$b:B$
is
terminal
if for any
$x:B$
the type
${\mathsf {hom}}_B(x,b)$
is contractible. That is, if the type
\begin{align*} {\mathsf {isterminal}}(b):\equiv \prod \limits _{x:B}{\mathsf {iscontr}}({\mathsf {hom}}_B(x,b)) \end{align*}
is inhabited.
In a Segal type, initial and terminal elements are unique up to isomorphism.
Proposition 2.27.
Let
$A$
be a Segal type. The following is true:
-
1. If
$a:A$
is an initial element, then it is unique up to isomorphism.
-
2. If
$a:A$
is a terminal element, then it is unique up to isomorphism.
Proof.
Assume that
$A$
is a Segal type and let
$a,b:A$
be initial elements. The types
${\mathsf {hom}}_A(a,b),{\mathsf {hom}}_A(b,a)$
are contractible, thus inhabited, say by
$f,g$
, respectively. We get the terms
$f\circ g :{\mathsf {hom}}_A(b,b),g\circ f :{\mathsf {hom}}_A(a,a)$
and
${\mathsf {id}}_b:{\mathsf {hom}}_A(b,b), {\mathsf {id}}_a:{\mathsf {hom}}_A(a,a)$
. Again, these hom types are contractible so
$f\circ g={\mathsf {id}}_b$
and
$g\circ f={\mathsf {id}}_a$
. By Proposition 2.15,
$f$
is an isomorphism between
$a$
and
$b$
. Therefore, initial terms in Segal types are unique up to isomorphism. Similarly, a terminal element is unique up to isomorphism.
Proposition 2.28.
A covariant type family
$C : A\rightarrow \mathcal{U}$
over a Segal type
$A$
is representable if and only if there exists an initial element
$(a, u)$
in the type
$\sum \limits _{x:A}C(x)$
, in which case
\begin{align*} {\mathsf {yon}}_a^C(u):\prod \limits _{x:A}({\mathsf {hom}}_A(a,x)\rightarrow C(x)) \end{align*}
defines an equivalence.
Proof. This is [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 9.10].
The majority of the results we have cited are stated in terms of covariant families. However, for contravariant families, we have similar results. The proofs for these statements are completely analogous.
3. Limits and Colimits
There is a well-known path to define limits and colimits in
$\infty$
-categories, as in (Lurie Reference Lurie2009) for quasi-categories or in Segal spaces (Rasekh Reference Rasekh2023). The strategy is used is first establish an adequate notion of cones and cocones. The second step is to define initial and terminal objects. Then, a limit is a “terminal cone,” and a colimit is an “initial cocone.” We follow the same approach to define them in sHoTT using the machinery of covariant families.
For types
$A,B$
and
$b:B$
, define the constant function
\begin{equation*} \triangle b:\equiv \lambda a.b:A\rightarrow B. \end{equation*}
Remark 3.1. If
$B$
is a Segal type then by Theorem 2.10,
$B^A:\equiv A\rightarrow B$
is also a Segal type. Let
$f:A\rightarrow B$
, from Proposition 2.19 and the fact that covariance and contravariance are stable under substitution (precomposition), then the type families
are covariant and contravariant, respectively.
Definition 3.2.
Let
$f:A\rightarrow B$
be a function. Define the type of
cocones
of
$f$
as the type
\begin{equation*}{\mathsf {cocone}}(f):\equiv \sum \limits _{b:B}{\mathsf {hom}}_{B^A}(f,\triangle b),\end{equation*}
and the type of
cones
of
$f$
as the type
\begin{equation*}{\mathsf {cone}}(f):\equiv \sum \limits _{b:B}{\mathsf {hom}}_{B^A}(\triangle b,f).\end{equation*}
Observation 3.3. From Remark 2.12 , it follows that
\begin{equation*} {\mathsf {cocone}}(f)\simeq \sum \limits _{b:B}\prod \limits _{a:A}{\mathsf {hom}}_B(fa,b) \end{equation*}
and
\begin{equation*} {\mathsf {cone}}(f)\simeq \sum \limits _{b:B}\prod \limits _{a:A}{\mathsf {hom}}_B(b,fa). \end{equation*}
We can recognize the type of cones and cocones of a function as direct analogs to cones and cocones in category theory. In Subsection 6.1, we will verify how the definition above and the following are consistent with the semantics.
Definition 3.4.
A
colimit
for
$f:A\rightarrow B$
is an initial term of the type:
\begin{equation*} {\mathsf {cocone}}(f)\equiv \sum \limits _{x:B}{\mathsf {hom}}_{B^A}(f,\triangle x). \end{equation*}
A
limit
for
$f:A\rightarrow B$
is a terminal term of the type:
\begin{equation*} {\mathsf {cone}}(f)\equiv \sum \limits _{x:B}{\mathsf {hom}}_{B^A}(\triangle x,f) \end{equation*}
Remark 3.5. Since the total space of a covariant family over a Segal type is a Segal type, Remark 3.1 implies that the type of cocones of a function
$f:A \to B$
is a Segal type. Similarly, we also get that the type of cones is a Segal type. We will make use of these two facts where needed without explicit mention.
We can show the following:
Corollary 3.6.
Let
$B$
be a Segal type and
$f:A\rightarrow B$
a function. Limits and colimits are unique up to a unique isomorphism if they exist.
We will obtain in Corollary 4.6 uniqueness up to equality property under the additional assumption that
$ B$
is a Rezk type. The following characterization is more flexible in terms of computations. We can think of it as the universal property of limits and colimits.
Proposition 3.7.
Let
$B$
a Segal type and
$f:A\rightarrow B$
be a function. There exists a colimit
$(b,\beta )$
for
$f$
if and only if
\begin{align*} \prod \limits _{x:B}({\mathsf {hom}}_B(b,x)\simeq {\mathsf {hom}}_{B^A}(f,\triangle x)). \end{align*}
Similarly, there exists a limit
$(a,\alpha )$
for
$f$
if and only if
\begin{align*} \prod \limits _{x:B}({\mathsf {hom}}_B(x,a)\simeq {\mathsf {hom}}_{B^A}(\triangle x,f)). \end{align*}
Proof.
The existence of a colimit
$(b,\beta )$
for the function
$f$
, that is ,
$(b,\beta )$
is an initial term of the type
$\sum \limits _{x:B}{\mathsf {hom}}_{B^A}(f,\triangle x)$
. By Proposition 2.28 is equivalent to say that the covariant family
\begin{align*} {\mathsf {hom}}_{B^A}(f,\triangle -):B\rightarrow \mathcal{U} \end{align*}
is representable. The proof for the second part is similar.
Remark 3.8. We can get an explicit description of the cone and cocones in Proposition 3.7. Under the same assumptions, given a family of equivalences
\begin{align*} \phi :\prod \limits _{x:B}({\mathsf {hom}}_B(x,a)\simeq {\mathsf {hom}}_{B^A}(\triangle x,f)), \end{align*}
we can take
$\phi _a: {\mathsf {hom}}_B(a,a) \simeq {\mathsf {hom}}_{B^A}(\triangle a, f)$
. This give us the cone
$\phi _a({\mathsf {id}}_a):{\mathsf {hom}}_{B^A}(\triangle a, f)$
. Following the usual categorical argument, we can show that
$(a,\phi _a({\mathsf {id}}_a))$
is the limit for
$f$
. Similarly, we can get the cocone for the apex
$b$
.
In the situation of Proposition 3.7, it is common practice to say that “
$b$
” is the colimit and that “
$a$
” is the limit of
$f:A\to B$
, respectively. This leaves implicit the data of the cocone and the cone, respectively.
3.1 Preservation of limits and colimits
The main goal of this section is to reproduce the classical result that limits are preserved under right adjoints. The same argument will show that left adjoints preserve colimits. Different presentations of adjunctions are studied in (Riehl and Shulman Reference Riehl and Shulman2017), namely, transposing and diagrammatic adjunctions, and it is shown that they are equivalent. The first kind of adjunction corresponds to the standard definition of adjunction in category theory in terms of hom sets, whereas the second resembles the triangle identities. For us, it will be enough to know that a quasi-transposing adjunction between types
$A,B$
consists of functions
$f:A\rightarrow B$
and
$u:B\rightarrow A$
and a family of maps
\begin{equation*} \phi :\prod \limits _{a:A,b:B}{\mathsf {hom}}_B(fa,b)\rightarrow {\mathsf {hom}}_A(a,ub) \end{equation*}
with quasi-inverses given by the maps
\begin{equation*} \psi :\prod \limits _{a:A,b:B}{\mathsf {hom}}_A(a,ub)\rightarrow {\mathsf {hom}}_B(fa,b) \end{equation*}
and homotopies
\begin{equation*}\xi :\prod _{a,b,k}\phi _{a,b}(\psi _{a,b}(k))=k,\quad \zeta :\prod _{a,b,l}\psi _{a,b}(\phi _{a,b}(l))=l.\end{equation*}
In this situation,
$u$
together with the data above is a quasi-transposing right adjoint for
$f$
. Using this notation, we have:
Theorem 3.9.
Let
$A,B$
be Segal types and functions
$g:J\rightarrow B$
,
$f:A\rightarrow B$
,
$u:B\rightarrow A$
. If
$g$
has a limit
$(b,\beta )$
, and
$u$
is a quasi-transposing right adjoint of
$f$
, then
$(u(b),u\beta )$
is a limit for
$ug:J\rightarrow A$
.
Proof. In view of Proposition 3.7, we have
\begin{equation*} \gamma :\prod _{x:B}({\mathsf {hom}}_B(x,b) \simeq {\mathsf {hom}}_{B^J}(\triangle x,g)), \end{equation*}
and we only need to show that
\begin{equation*} \prod _{y:A}({\mathsf {hom}}_A(y,u(b)) \simeq {\mathsf {hom}}_{A^J}(\triangle y,ug)). \end{equation*}
Recall that
${\mathsf {hom}}_{B^J}(\triangle x,g) \simeq \prod _{j:J} {\mathsf {hom}}_B(x,g(j))$
. Then, for any
$y:A$
, we get
\begin{equation*}\gamma _{f(y)}: {\mathsf {hom}}_B(f(y),b) \simeq \prod _{j:J} {\mathsf {hom}}_B(f(y),g(j)). \end{equation*}
Using the adjunction, we get a chain of equivalences:
\begin{align*} {\mathsf {hom}}_A(y,u(b)) &\simeq {\mathsf {hom}}_B(f(y),b) \\ &\simeq \prod _{j:J} {\mathsf {hom}}_B(f(y),g(j)) \\ &\simeq \prod _{j:J} {\mathsf {hom}}_A(y,ug(j)) \\ &\simeq {\mathsf {hom}}_{A^J}(\triangle y,ug). \end{align*}
This gives us the family of equivalences
\begin{equation*} \Gamma : \prod _{y:A}({\mathsf {hom}}_A(y,u(b)) \simeq {\mathsf {hom}}_{A^J}(\triangle y,ug)). \end{equation*}
To complete the proof, note that the cone for
$u(b)$
can be obtained from
$\beta :{\mathsf {hom}}_{B^J}(\triangle b,g)$
, since
$u\beta :{\mathsf {hom}}_{A^J}(\triangle u(b),ug)$
. Following Remark 3.8, we must have that
$\Gamma _{u(b)}({\mathsf {id}}_{u(b)})= u\beta$
.
We also have:
Corollary 3.10.
If
$A,B$
are Segal types and
$f:A\rightarrow B$
has a left quasi-transposing adjoint, then
$f$
preserves colimits of functions
$g:J\rightarrow A$
.
Proof. Similar to the previous theorem.
4. (Co)Limits in Rezk Types
In this section, we restrict our attention to colimits in a Rezk type, which allows us to improve some of our results.
In what follows, for
$C:A\rightarrow \mathcal{U}$
a type family, we denote by
\begin{equation*}\tilde {C} :\equiv \sum \limits _{x:A}C(x)\end{equation*}
the dependent sum over
$A$
.
Lemma 4.1.
Assume that
$C:A\rightarrow \mathcal{U}$
is a covariant family over a Segal type
$A$
, let
$(a,u),(b,v):\sum \limits _{x:A}C(x)$
. Then,
\begin{equation*} ((a,u)\cong (b,v))\simeq \sum \limits _{f:a\cong b}f_*u=v. \end{equation*}
Proof. First, from Theorem 2.2, we have the equivalence
\begin{align*} {\mathsf {hom}}_{\tilde {C}}((a,u),(b,v))\simeq \sum \limits _{f:{\mathsf {hom}}_A(a,b)}{\mathsf {hom}}_{C(f)}(u,v). \end{align*}
Tracing through this equivalence, we get the projection
$\pi _1:\tilde {C} \rightarrow A$
. Since
$C:A\rightarrow \mathcal{U}$
is a covariant family over the Segal type
$A$
,
$\tilde {C}$
is a Segal type [(Riehl and Shulman Reference Riehl and Shulman2017), Theorem 8.8]. By Proposition 2.11, the projection respects identities and composition. In particular, if
$\bar {f}:(a,u)\cong (b,v)$
then
$\pi _1(\bar {f}):a\cong b$
. Furthermore, covariance of
$C$
implies that
$(v,\pi _2(\bar {f}))=(\pi _1(\bar {f})_*u,{\mathsf {trans}}_{\pi _1(\bar {f}),u})$
, hence,
$ p_{\bar {f}}:\pi _1(\bar {f})_*u=v$
. Therefore, we can define
\begin{equation*} \phi :((a,u)\cong (b,v))\rightarrow \sum \limits _{f:a\cong b}f_*u=v \end{equation*}
to be
$\phi :\equiv \lambda \bar {f}.(\pi _1(\bar {f}),p_{\bar {f}})$
where
$p_{\bar {f}}:\pi _1(\bar {f})_*u=v$
.
In order to construct the function in the other direction, we observe the following: if
$f:a\cong b$
, then, there exist
$g:{\mathsf {hom}}_A(b,a)$
with
$f\circ g={\mathsf {id}}_b$
and
$g\circ f={\mathsf {id}}_a$
. Hence, covariance of
$C$
implies that the types
\begin{equation*} \sum \limits _{w: C(b)}{\mathsf {hom}}_{C(f)}(u,w) \text{ and } \sum \limits _{w: C(a)}{\mathsf {hom}}_{C(g)}(v,w) \end{equation*}
are contractible with centers of contraction
$(f_*u,{\mathsf {trans}}_{f,u})$
and
$(g_*v,{\mathsf {trans}}_{g,v})$
, respectively. Note also that
\begin{equation*} u=({\mathsf {id}}_a)_*u=(g\circ f)_*u=g_*(f_*u)=g_*v. \end{equation*}
We get elements
\begin{equation*} (f,{\mathsf {trans}}_{f,u}):\sum \limits _{h:{\mathsf {hom}}_A(a,b)}{\mathsf {hom}}_{C(h)}(u,v) \end{equation*}
and
\begin{equation*} (g,{\mathsf {trans}}_{g,v}):\sum \limits _{h:{\mathsf {hom}}_B(b,a)}{\mathsf {hom}}_{C(h)}(v,u). \end{equation*}
Using the equivalence
\begin{align} \left (\sum \limits _{k: {\mathsf {hom}}_A(x,y)}{\mathsf {hom}}_{C(k)}(w,z)\right )\simeq {\mathsf {hom}}_{\tilde {C}}((x,w),(y,z)) \end{align}
we get
$\bar {f}:{\mathsf {hom}}_{\tilde {C}}((a,u),(b,v))$
and
$\bar {g}:{\mathsf {hom}}_{\tilde {C}}((b,v),(a,u))$
. Since
$\tilde {C}$
is a Segal type then
$\bar {g}\circ \bar {f}: {\mathsf {hom}}_{\tilde {C}}((a,u),(a,u))$
. We have the composition in
$A$
and the dependent composition, so we obtain the 2-simplex
and
We get a term
$\big (\overline {g\circ f},({\mathsf {comp}}_{g,f},{\mathsf {comp}}_{{\mathsf {trans}}_{g,v},{\mathsf {trans}}_{f,u}})\big )$
in
Since this type is contractible, in particular, we have
$\bar {g}\circ \bar {f}=\overline {g\circ f}$
. Under the equivalence ( 1),
${\mathsf {id}}_a$
is mapped to
${\mathsf {id}}_{(a,u)}$
, so
\begin{equation*}\bar {g}\circ \bar {f}=\overline {g\circ f}=\overline {{\mathsf {id}}}_a={\mathsf {id}}_{(a,u)}.\end{equation*}
Similarly,
$\bar {f}\circ \bar {g}={\mathsf {id}}_{(b,v)}$
. Therefore, we define
\begin{equation*} \psi :\sum \limits _{f:a\cong b}f_*u=v\rightarrow ((a,u)\cong (b,v)) \end{equation*}
as
$\psi :\equiv \lambda (f,p).\bar {f}$
. The functions
$\phi$
and
$\psi$
give us the required equivalence.
A related result is [(The Univalent Foundations Program 2013), Theorem 2.7.2], instead of identity types, we now have isomorphism types. Thus, an isomorphism in
$\Sigma$
-types is determined by an isomorphism in the base together with a dependent isomorphism in the total type lying over it. We formulate the result but give no proof corresponding to contravariant families.
Lemma 4.2.
Assume that
$C:A\rightarrow \mathcal{U}$
is a contravariant family over a Segal type
$A$
, let
$(a,u),(b,v):\sum \limits _{x:A}C(x)$
. Then,
\begin{equation*} ((a,u)\cong (b,v))\simeq \sum \limits _{f:a\cong b}u=f^*v. \end{equation*}
Theorem 4.3.
If
$A$
is a Rezk type and
$C:A\rightarrow \mathcal{U}$
is a covariant (contravariant) family over
$A$
, then
$\sum \limits _{x:A}C(x)$
is a Rezk type.
Proof.
Let
$(a,u),(b,v):\sum \limits _{x:A}C(x)$
. We have
\begin{align*} \big ((a,u)=(b,v)\big )\simeq & \sum \limits _{f:a=b}{\mathsf {transport}}^C(f,u)=v \\ \simeq & \sum \limits _{{\mathsf {idtoiso}}(f):a\cong b}{\mathsf {idtoiso}}(f)_*u=v \\ \simeq & (a,u)\cong (b,v). \end{align*}
The equivalence in the middle follows from
$A$
being Rezk type and from Lemma 2.22 where we have the equality
${\mathsf {idtoiso}}(e)_* u = {\mathsf {transport}}^C (e, u)$
. The function
\begin{equation*} {\mathsf {idtoiso}}:(a,u)=(b,v)\rightarrow (a,u)\cong (b,v) \end{equation*}
is exactly the equivalence above. Indeed, we can use path induction to see this. If
$(a,u)=(a,u)$
then
${\mathsf {refl}}_{(a,u)}$
is mapped to
$({\mathsf {refl}}_a,{\mathsf {refl}}_u)$
. This goes to
$({\mathsf {id}}_a,{\mathsf {refl}}_u)$
which finally corresponds to
${\mathsf {id}}_{(a,u)}:(a,u)\cong (a,u)$
. Therefore,
$\sum \limits _{x:A}C(x)$
is Rezk.
Remark 4.4. Theorem 4.3 can also be deduced from Proposition 4.2.6 and Corollary 6.1.4 of (Buchholtz and Weinberger Reference Buchholtz and Weinberger2023). Their proof makes use of their theory of (co)cartesian fibrations in sHoTT that they develop throughout their work. Ours is more elementary and simply uses the characterization of the type of isomorphisms between elements in a
$\sum$
-type we provided in Lemma 4.1.
Definition 4.5.
Let
$B$
be a Segal type,
$f:A\rightarrow B$
a function. We define the type:
\begin{equation*} {\mathsf {hasColimit}}(f):\equiv \sum \limits _{w:{\mathsf {cocone}}(f)}{\mathsf {isinitial}}(w) \end{equation*}
and also
\begin{equation*} {\mathsf {hasLimit}}(f):\equiv \sum \limits _{w:{\mathsf {cone}}(f)}{\mathsf {isterminal}}(w). \end{equation*}
In Corollary 3.6, we showed that limits and colimits are unique up to isomorphism. Now, we can show further:
Corollary 4.6.
If
$B$
is a Rezk type and we have a function
$f:A\rightarrow B$
, then, the types
${\mathsf {hasColimit}}(f)$
and
${\mathsf {hasLimit}}(f)$
are propositions.
Proof.
Let
$(a,\alpha ), (b,\beta )$
be limits for
$f$
, by Corollary 3.6, we have that
$(a,\alpha )\cong (b,\beta )$
. Theorem 4.3 applied to the covariant family
\begin{equation*} {\mathsf {hom}}_{B^A}(f,\triangle -):B\rightarrow \mathcal{U} \end{equation*}
implies that
${\mathsf {cone}}(f)$
is also a Rezk type. Therefore,
\begin{equation*} \big ((a,\alpha )= (b,\beta )\big )\simeq \big ((a,\alpha )\cong (b,\beta )\big ) \end{equation*}
hence,
${\mathsf {hasLimit}}(f)$
is a proposition. A similar proof shows that
${\mathsf {hasColimit}}(f)$
is also a proposition.
5. Limit of Spaces as Dependent Product
The goal of this section is to replicate (in sHoTT) the following result which is known to be true in presentations for
$\infty$
-categories. Let
$\{G_i\}_{i\in I}$
be a family of
$\infty$
-groupoids indexed by a set
$I$
. Denote by
$G$
the obvious diagram
$I \to \infty \text{-}\textbf {Gpd}$
, then, we have that
\begin{equation*} \lim _I\, G_i=\prod _{i\in I}G_i. \end{equation*}
This result is known to be true in models such as complete Segal spaces or quasi-categories.
It is natural to expect that a similar result is true in sHoTT. However, such computation is not straightforward: The obstruction is that in our type theory, we do not have an
$\infty$
-category of
$\infty$
-groupoids in which we can take this limit. The way to address this issue is by using the notion of univalent covariant families originally due to Cavallo-Riehl-Sattler (Cavallo et al. Reference Cavallo, Riehl and Sattler2018) to sufficiently axiomatize the properties of the
$\infty$
-category of spaces.
Let
$B$
be a Segal type. For any covariant family
$E:B\rightarrow \mathcal{U}$
and
$a,b:B$
, using the notation of Definition 2.18, we obtain a function
\begin{equation*} {\mathsf {arrtofun}}:{\mathsf {hom}}_B(a,b)\rightarrow \big (E(a)\rightarrow E(b)\big ) \end{equation*}
where for any
$f: {\mathsf {hom}}_B(a,b)$
, we set
\begin{equation*} {\mathsf {arrtofun}}(f):\equiv f_*\equiv \lambda u.f_*u. \end{equation*}
Observation 5.1.
From Proposition
2.20
, for any
$f : {\mathsf {hom}}_B(x, y)$
,
$g : {\mathsf {hom}}_B(y, z)$
, and
$u : E(x)$
, we have
$g_* (f_* u) = (g\circ f )_* u$
. As a consequence, we have
${\mathsf {arrtofun}}(g\circ f)={\mathsf {arrtofun}}(g){\mathsf {arrtofun}}(f)$
.
Definition 5.2.
Let
$E:B\rightarrow \mathcal{U}$
be a covariant family over a Segal type
$B$
. We say that
$E$
is
directed univalent
if for all
$a,b:B$
the map
$\mathsf {arrtofun}$
is an equivalence. Denote its inverse by
\begin{equation*} {\mathsf {dua}}:\big (E(a)\rightarrow E(b)\big )\rightarrow {\mathsf {hom}}_B(a,b). \end{equation*}
We can understand the family
$E$
as satisfying a “directed univalent axiom.” The prime example of a directed univalent fibration in the semantics should be the generic fibration associated to the
$\infty$
-category of
$\infty$
-groupoids. In [(Riehl and Shulman Reference Riehl and Shulman2017), Remark A.27], it is shown that covariant families correspond to left fibrations. Cavallo, Riehl, and Sattler announced in (Cavallo et al. Reference Cavallo, Riehl and Sattler2018), the existence of a fibrant universe of left fibrations in the simplicial spaces model. This implies the existence of an instance of Definition 5.2, see also Remark 5.15.
The result in [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 8.18] shows that if
$ E:B \to \mathcal{U}$
is a covariant type family over a Segal type
$B$
, then for any
$b:B$
the type
$E(b)$
is discrete. Furthermore, if
$E$
is directed univalent, we could think of
$B$
as a Segal type “that as has terms discrete types.” We will make this idea precise in Corollary 5.9. In this part of the paper, we use the extensionality axiom and the notation introduced in Subsection 2.1.1. Recall that
$f\sim g$
expresses the fact that two functions
$f,g:X \to Y$
are homotopic.
Lemma 5.3.
Let
$B$
a Segal type and
$E:B\rightarrow \mathcal{U}$
a directed univalent covariant family and
$a,b:B$
. If
$\delta :E(a)\rightarrow E(b)$
is an equivalence, then
${\mathsf {dua}}(\delta )$
is an isomorphism.
Proof.
By Proposition 2.20 for any
$x:B$
and
$u:E(x)$
, we have
$({\mathsf {id}}_{x})_*(u)=u$
. Therefore,
${\mathsf {arrtofun}}({\mathsf {id}}_x)={\mathsf {id}}_{E(x)}$
. This also implies that
\begin{equation*} {\mathsf {dua}}({\mathsf {id}}_{E(x)})={\mathsf {dua}}({\mathsf {arrtofun}}({\mathsf {id}}_x))={\mathsf {id}}_x. \end{equation*}
Treat
$\delta :E(a)\rightarrow E(b)$
as a bi-invertible function, so there is
$\gamma :E(b)\rightarrow E(a)$
such that
$\delta \gamma \sim {\mathsf {id}}_{E(b)}$
and
$\gamma \delta \sim {\mathsf {id}}_{E(a)}$
. From the extensionality axiom, we can further assume
$\delta \gamma ={\mathsf {id}}_{E(b)}$
and
$\gamma \delta ={\mathsf {id}}_{E(a)}$
. Hence,
${\mathsf {dua}}(\delta \gamma )={\mathsf {id}}_b$
and
${\mathsf {dua}}(\gamma \delta )={\mathsf {id}}_a$
. By Observation 5.1, we have that
\begin{align*} {\mathsf {arrtofun}}({\mathsf {dua}}(\delta )\circ {\mathsf {dua}}(\gamma )) &={\mathsf {arrtofun}}({\mathsf {dua}}(\delta )){\mathsf {arrtofun}}({\mathsf {dua}}(\gamma ))) \\ &= \delta \gamma \\ &={\mathsf {id}}_{E(b)}. \end{align*}
It follows that
${\mathsf {dua}}(\delta )\circ {\mathsf {dua}}(\gamma )={\mathsf {dua}}(\delta \gamma )={\mathsf {id}}_b$
. Similarly,
\begin{equation*} {\mathsf {dua}}(\gamma )\circ {\mathsf {dua}}(\delta )={\mathsf {dua}}(\gamma \delta )={\mathsf {id}}_a,\end{equation*}
which concludes the proof of the statement.
Definition 5.4.
Given a type
$A$
and a family
$E:B\rightarrow \mathcal{U}$
over a type
$B$
, we define the type
\begin{equation*} {\mathsf {issmall}}^E_B(A):\equiv \sum \limits _{b:B}\big (E(b)\simeq A\big ). \end{equation*}
We say that
$A$
is
$E$
-
small
if
${\mathsf {issmall}}^E_B(A)$
is inhabited.
A related notion appears in [(Rijke Reference Rijke2025), Definition 17.1.3] where they define what it means for a type
$A$
to be small with respect to a univalent universe
$\mathcal{U}$
; a type
$A$
is said to be
$\mathcal{U}$
-small if there is type
$X:\mathcal{U}$
and an equivalence between
$A$
and
$X$
. The same definition in ibid appears in [(Bezem et al. Reference Bezem, Buchholtz, Cagne, Dundas and Grayson2025), Section 2.19] under the name essentially
$\mathcal{U}$
-small.
Remark 5.5. In the rest of the paper, we use Definition 5.4 under the assumption that the family
$E:B \to \mathcal{U}$
is covariant. One could also think of assuming that
$E$
is contravariant, but we have no use for it.
Remark 5.6. Since we will require a covariant family, we can express whether a discrete type
$A:\mathcal{U}$
is
$B$
-small. First, [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 8.18] proves that for a covariant type family over a Segal type
$E:B \to \mathcal{U}$
each
$E(b)$
is discrete. Therefore, if we have a proof of
${\mathsf {issmall}}^E_B(A)$
, then
$A$
is also discrete since it is equivalent to a discrete type, and of course it is
$B$
-small.
A more general and concrete example of this can be seen in the semantics. Given a Grothendieck universe
$U$
, one could build a sub-universe
$U'$
(an
$\infty$
-category) that consists of
$\kappa$
-small
$\infty$
-groupoids, where
$\kappa$
is a regular cardinal. By construction, elements of
$U'$
are exactly the
$U'$
-small elements of
$U$
. Of course, this would be one instance of Definition 5.2.
The idea of Definition 5.4 is that
$B$
has an element representing the type
$A$
. We can think of
$A$
as “a type in
$B$
,” hence, the name
$B$
-small. To make sense of this, we need to note some simple facts. The first lemma simply computes
${\mathsf {transport}}^D$
for the type family
$D:B \to \mathcal{U}$
as defined below.
Lemma 5.7.
Let
$E:B\rightarrow \mathcal{U}$
be a directed univalent covariant family over a Rezk type
$B$
. For a type
$A$
, define
$D:B\rightarrow \mathcal{U}$
as
$D(b):\equiv E(b)\simeq A$
. Then, for any
$b,b':B$
,
$p:b=b'$
and
$\sigma :E(b)\simeq A$
\begin{equation*} {\mathsf {transport}}^D(p,\sigma )=\sigma ({\mathsf {arrtofun}}({\mathsf {idtoiso}}(p^{-1}))). \end{equation*}
Proof.
By path induction, we can assume that
$p\equiv {\mathsf {refl}}_b:b=b$
. In this case,
\begin{equation*} {\mathsf {transport}}^D({\mathsf {refl}}_b,\sigma )\equiv {\mathsf {id}}_{D(b)}(\sigma )=\sigma . \end{equation*}
On the right-hand side, we have
\begin{align*} \sigma ({\mathsf {arrtofun}}({\mathsf {idtoiso}}(p^{-1}))) &\equiv \sigma ({\mathsf {arrtofun}}({\mathsf {idtoiso}}({\mathsf {refl}}_b))) \\ &\equiv \sigma ({\mathsf {arrtofun}}({\mathsf {id}}_b))\\ &\equiv \sigma {\mathsf {id}}_{E(b)} \\ &=\sigma \end{align*}
hence, we can simply use
${\mathsf {refl}}_{\sigma }:\sigma =\sigma$
.
Remark 5.8. Under the same assumptions of the previous lemma, we also have by path induction that for any
$p:b=b',$
\begin{equation*} {\mathsf {arrtofun}}({\mathsf {idtoiso}}(p^{-1}))={\mathsf {arrtofun}}({\mathsf {idtoiso}}(p))^{-1} \end{equation*}
since if
$p\equiv {\mathsf {refl}}_b$
then both sides of the equality are
${\mathsf {id}}_{E(b)}$
. Here,
${\mathsf {arrtofun}}({\mathsf {idtoiso}}(p))^{-1}$
abusively denotes an inverse for
${\mathsf {arrtofun}}({\mathsf {idtoiso}}(p))$
.
The two results above have the following consequence:
Corollary 5.9.
Let
$E:B\rightarrow \mathcal{U}$
be a directed univalent covariant family over a Rezk type
$B$
. Then, for any type
$A$
, the type
${\mathsf {issmall}}^E_B(A)$
is a proposition.
Proof.
Let
$(b,\sigma ),(b',\sigma '):\sum \limits _{b:B}\big (E(b)\simeq A\big )$
. We have
\begin{equation*} \big ((b,\sigma )=(b',\sigma ')\big )\simeq \sum \limits _{p:b=b'}\sigma =^p\sigma ' \end{equation*}
therefore is enough to give a term of the right-hand side. We treat
$\sigma$
and
$\sigma '$
as bi-invertible functions. Let
$\delta :A\rightarrow E(b)$
and
$\delta ':A\rightarrow E(b')$
be their inverses, respectively. We get an equivalence
$\delta '\sigma :E(b)\simeq E(b')$
, and by Lemma 5.3
${\mathsf {dua}}(\delta '\sigma ):b\cong b'$
. Since
$B$
is a Rezk type,
${\mathsf {rezk}}({\mathsf {dua}}(\delta '\sigma )):b=b'$
. From the Lemma 5.7 above and the fact that
$(({\mathsf {arrtofun}})({\mathsf {idtoiso}}))(({\mathsf {rezk}})({\mathsf {dua}}))={\mathsf {id}}_{E(b')^{E(b)}}$
we obtain:
\begin{align*} {\mathsf {transport}}^D({\mathsf {rezk}}({\mathsf {dua}}(\delta '\sigma )),\sigma ) &=\sigma ({\mathsf {arrtofun}}({\mathsf {idtoiso}}({\mathsf {rezk}}({\mathsf {dua}}(\delta '\sigma ))^{-1})) \\ &=\sigma ({\mathsf {arrtofun}}({\mathsf {idtoiso}}({\mathsf {rezk}}({\mathsf {dua}}(\delta '\sigma ))))^{-1}) \\ &=\sigma (\delta '\sigma )^{-1} \\ &=\sigma (\delta \sigma ')\\ &={\mathsf {id}}_{A}\sigma ' \\ &=\sigma ' \end{align*}
where again
$(\delta '\sigma )^{-1}$
denotes an inverse for
$\delta '\sigma$
. Since
$\delta \sigma '$
is also an inverse, the two are equal.
Remark 5.10. In the situation of Corollary 5.9, if a type
$A$
is
$B$
-small then
${\mathsf {issmall}}^E_B(A)$
is contractible. This entails the existence of a center of contraction
$(b_0,\sigma _0)$
, where
$b_0$
is a term of type
$B$
and
$\sigma _0$
is an equivalence between
$E(b_0)$
and
$A$
. Hence,
$A$
is uniquely related to a term in
$B$
.
Therefore, given a directed univalent covariant family of types
$E:B\rightarrow \mathcal{U}$
, one can think of
$B$
as being the
$\infty$
-category of “small spaces,” or at least a full subcategory of it. However, note that we do not assume that
$B$
is closed under limits. The next proposition shows that as long as a dependent product of small types is itself small, it corresponds to a limit in
$B$
.
Proposition 5.11.
Let
$B$
a Rezk type, a function
$f:I\rightarrow B$
and assume that
$E:B\rightarrow \mathcal{U}$
is a directed univalent covariant family such that
$\prod \limits _{i:I}E(f(i))$
is
$B$
-small. If
$(b_0,\sigma _0)$
is the center of contraction of
${\mathsf {issmall}}^E_B(\prod \limits _{i:I}E(f(i)))$
, then
$b_0$
is the limit for the function
$f$
.
Proof.
Indeed, using Proposition 3.7, it is enough to show that for all
$b:B$
\begin{equation*} {\mathsf {hom}}_B(b,b_0)\simeq {\mathsf {hom}}_{B^I}(\triangle b,f). \end{equation*}
For the right-hand side, we have
\begin{equation*} {\mathsf {hom}}_{B^I}(\triangle b,f)\simeq \left ( \prod \limits _{i:I}{\mathsf {hom}}_{B}(b,f(i))\right ) \simeq \left (\prod \limits _{i:I}E(b)\rightarrow E(f(i)) \right ) \end{equation*}
The last equivalence follows because
$E$
is directed univalent. For the same reason together with the assumption
$\sigma _0:E(b_0)\simeq \prod \limits _{i:I}E(f(i))$
, the left-hand side is equivalent to
\begin{equation*} \big (E(b)\rightarrow E(b_0)\big ) \simeq \left (E(b)\rightarrow \prod \limits _{i:I}E(f(i))\right ). \end{equation*}
Moreover, we have the equivalence
\begin{equation*} \left (E(b)\rightarrow \prod \limits _{i:I}E(f(i))\right )\simeq \left (\prod \limits _{i:I}E(b)\rightarrow E(f(i))\right ). \end{equation*}
For the rest of the section, we want to show that the condition over the dependent product of the fiber being
$B$
-small in Proposition 5.11 is necessary for the existence of the limit.
Observation 5.12.
Suppose that
$D:B\rightarrow \mathcal{U}$
is a constant covariant family at
$D:\mathcal{U}$
over a Segal type
$B$
. Take any
$f:{\mathsf {hom}}_B(x,y)$
and
$w:D$
, then, the type
\begin{equation*}\sum \limits _{v:D}{\mathsf {hom}}_{D(f)}(w,v)\end{equation*}
has center of contraction
$(f_*w,{\mathsf {trans}}_{f,w})$
. The point
$(w,\lambda (t: \unicode{x1D7DA}).w)$
belongs to the aforementioned type, which therefore must be equal to the center of contraction. In particular, this implies that
$f_*w=w$
.
This observation has the following consequence:
Lemma 5.13.
Let
$B:\mathcal{U}$
a Segal type and
$E:B\rightarrow \mathcal{U}$
a directed univalent covariant family over
$B$
. If there is
$u:E(b_0)$
for some
$b_0:B$
, then for all
$b_1:B$
\begin{equation*} E(b_1)\simeq \prod \limits _{b:B}{\mathsf {hom}}_B(b,b_1).\end{equation*}
Proof.
The covariant family
$E:B\rightarrow \mathcal{U}$
is directed univalent, so instead we may prove that for all
$b_1:B$
\begin{equation*} E(b_1)\simeq \left (\prod \limits _{b:B}E(b)\rightarrow E(b_1)\right ).\end{equation*}
Let
$b_1:B$
and define
\begin{equation*} F:E(b_1)\rightarrow \left (\prod \limits _{b:B}E(b)\rightarrow E(b_1)\right ),\end{equation*}
where
$F(v):\equiv F_v$
is the dependent function
$\lambda (b:B).\lambda (w:E(b)).v$
. The function in the other direction
\begin{equation*} G:\left (\prod \limits _{b:B}E(b)\rightarrow E(b_1)\right )\rightarrow E(b_1) \end{equation*}
for each
$\phi :\left (\prod \limits _{b:B}E(b)\rightarrow E(b_1)\right )$
is defined by
$G(\phi ):\equiv \phi _{b_0}(u)$
. It is immediate that
\begin{equation*} G(F(v))\equiv G(F_v)\equiv (F_v)_{b_0}(u)\equiv v,\end{equation*}
so the composition is the identity. For any
$\phi :(\prod \limits _{b:B}E(b)\rightarrow E(b_1))$
, we have:
$F(G(\phi ))\equiv F(\phi _{b_0}(u))\equiv F_{\phi _{b_0}(u)}$
. We need to show that this dependent function is equal to
$\phi$
. Consider the family
\begin{equation*}D:B\rightarrow \mathcal{U}\end{equation*}
$D(b):\equiv E(b_1)$
; this defines a covariant family. For any
$b:B$
define
$f:E(b)\rightarrow E(b_0)$
as
$f(w):\equiv u$
. Since the family
$E$
is directed univalent,
${\mathsf {dua}}(f):{\mathsf {hom}}_B(b,b_0)$
and
\begin{align} {\mathsf {dua}}(f)_*\equiv {\mathsf {arrtofun}}({\mathsf {dua}}(f))=f \end{align}
where the definitional equality follows by definition of
$\mathsf {arrtofun}$
.
By naturality, as in Proposition 2.21, applied to the function
$\phi :\prod \limits _{b:B}E(b)\rightarrow D(b)$
and the arrow
${\mathsf {dua}}(f):{\mathsf {hom}}_B(b,b_0)$
, we have the following commutative square
where the horizontal functions are given by the specified type families. We now evaluate at any
$w:E(b)$
. Since
$D$
is constant over
$B$
, from Observation 5.12, we get the first equality below, the second follows by naturality, and the third equality comes from ( 2),
\begin{align*} \phi _b(w) &= {\mathsf {dua}}(f)_*(\phi _b(w)) \\ &= \phi _{b_0}({\mathsf {dua}}(f)_*(w)) \\ &= \phi _{b_0}(f(w)) \\ &= \phi _{b_0}(u) \\ &\equiv (F_{\phi _{b_0}(u)})_b(w), \end{align*}
while the second-to-last and last equality follow by definition of
$f$
and
$F$
, respectively. This establishes the equality
$\phi =F_{\phi _{b_0}(u)}$
. Therefore,
$FG$
is the identity on
$\prod \limits _{b:B}(E(b)\rightarrow E(b_1))$
, hence, the equivalence.
Finally, we have:
Proposition 5.14.
Let
$B$
a Rezk type and a function
$f:I\rightarrow B$
with limit
$(b_1,\alpha )$
. Assume that
$E:B\rightarrow \mathcal{U}$
is a directed univalent covariant family such that for some
$b_0:B$
there is
$u:E(b_0)$
. Then,
$E(b_1)\simeq \prod \limits _{i:I}E(f(i))$
.
This just means that as long as the family
$E:B\rightarrow \mathcal{U}$
has at least one fiber that is inhabited, then, the sufficient condition of Proposition 5.11 is also necessary for the existence of a limit.
Proof. From Lemma 5.13, we have
\begin{equation*}E(b_1)\simeq \prod \limits _{b:B}{\mathsf {hom}}_B(b,b_1),\end{equation*}
since
$b_1$
is the limit:
\begin{equation*}{\mathsf {hom}}_B(b,b_1)\simeq \prod \limits _{i:I}{\mathsf {hom}}_B(b,f(i)).\end{equation*}
Combining this equivalence, and using Lemma 5.13 again gives us:
\begin{align*} E(b_1) & \simeq \prod \limits _{b:B}\prod \limits _{i:I}{\mathsf {hom}}_B(b,f(i)) \\ & \simeq \prod \limits _{i:I}\prod \limits _{b:B}{\mathsf {hom}}_B(b,f(i)) \\ & \simeq \prod \limits _{i:I} E(f(i)). \end{align*}
Remark 5.15. In a recent paper (Gratzer et al. Reference Gratzer, Weinberger and Buchholtz2024), the authors construct the
$(\infty ,1)$
-category of
$\infty$
-groupoids. They achieve this by enhancing sHoTT with modalities. The type
$\mathcal{S}$
they construct satisfies exactly Definition 5.2. Our results in this section simply say that a diagram with values in
$\mathcal{S}$
has a limit in
$\mathcal{S}$
. This limit is given by the dependent product. We could rewrite our results by replacing our type
$B$
by
$\mathcal S$
, and each type
$E(b)$
for
$b:B$
by an element of
$\mathcal S$
. However, we keep our presentation as it is since we think it showcases a “resizing” technique, by means of Definition 5.4, that could be useful in other contexts.
6. The Bisimplicial Sets Semantics of sHoTT
In this section, we verify that our synthetic definitions of limit a colimit are semantically correct. We use the fact the semantics of sHoTT is the category of bisimplicial sets
$ \mathbf{ssSet}.$
In [(Riehl and Shulman, Reference Riehl and Shulman2017), Appendix A], it is shown that sHoTT can be interpreted in the category
$ \mathbf{ssSet}$
. Let us first recall the following result from (Shulman, Reference Shulman2015):
Theorem 6.1.
For any elegant Reedy category
$\mathcal{C}$
, the Reedy model structure on
$\mathbf{sSet}^{\mathcal{C}^{op}}$
supports a model of intensional type theory with dependent sums and products, identity types, and as many univalent universes as there are inaccessible cardinals greater than
$|\mathcal{C}|$
, where
$|\mathcal{C}|$
is the cardinal of the set of objects of the category
$\mathcal{C}$
.
Let us first review how to interpret some usual types. The one-point type is the easiest; it agrees with the terminal object in our model. According to (Shulman Reference Shulman2015), we can take a category
$\mathcal{M}$
that is locally cartesian closed and has a model structure that is simplicial, right proper, and the cofibrations are the monomorphisms. This is the situation of Theorem 6.1. If we have a map
$f:A\to B$
, then, the pullback functor
$f^*:\mathcal{M}/B \to \mathcal{M}/A$
has both a right and a left adjoint, which are denoted by
$\prod _f$
and
$\sum _f$
, respectively. Since fibrations are closed under compositions, the functor
$\sum _f$
preserves fibrations if
$f$
is itself a fibration. The functor
$f^*$
preserves cofibrations since these are exactly the monomorphisms, thus
$\prod _f$
preserving acyclic fibrations. Furthermore, one can also see that in combination with right properness
$\prod _f$
preserves fibrations, see [(Arndt and Kapulkin Reference Arndt and Kapulkin2011), Theorem 26]. This is how dependent sums and products are interpreted.
The Reedy model structure on bisimplicial sets gives rise to a comprehension category with shapes in which the judgment
$\Gamma \vdash A\,{\mathsf {type}}$
is directly interpreted as a fibration
$\Gamma \to A$
. Here, we are overlooking the fact that our types also may depend on topes and shapes. To see more details and the correct definitions, we refer the reader to our cited main reference (Riehl and Shulman Reference Riehl and Shulman2017). The technical result is:
Theorem 6.2. ([(Riehl and Shulman Reference Riehl and Shulman2017), Theorem A.18]) The comprehension category constructed from any model category with
$\mathfrak{T}$
-
shapes
has
pseudo-stable coherent tope logic
with type eliminations for tope disjunction and also
pseudo-stable extension types
satisfying relative function extensionality.
(Riehl and Shulman Reference Riehl and Shulman2017) uses the result on
$\mathfrak T$
the coherent theory of the strict interval of sHoTT presented in Subsection 2.1 or [(Riehl and Shulman Reference Riehl and Shulman2017), Section 3.1]. Furthermore, Weinberger shows in (Weinberger Reference Weinberger2022) that the semantical interpretation of extension types is strictly stable under pullbacks and not only pseudo-stable as in the theorem above. Moreover, [(Riehl and Shulman Reference Riehl and Shulman2017), A.3] proves:
Theorem 6.3.
Segal types correspond to Segal spaces in
$ \mathbf{ssSet}$
, and Rezk types to Rezk spaces (a.k.a. complete Segal spaces).
Remark 6.4. In Definition 2.14, we defined discrete types. However, the terminology might be confusing. According to [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 10.10] a type
$ A$
is discrete if and only if
$ A$
is Rezk and all its arrows are isomorphisms. On the other hand, [(Rezk Reference Rezk2001), Corollary 6.6] gives a similar characterization for complete Segal spaces, such simplicial spaces are called constant. A simplicial space
$ X$
is constant if for any map
$ [n] \to [m]$
, the induced map
$ X_m \to X_n$
is an equivalence of spaces. In particular, for all
$ n$
, we have
$ X_n=X_0$
. Hence, discrete types correspond to constant simplicial spaces. As a consequence, we interpret discrete types into the bisimplicial sets model as constant simplicial spaces. See also the discussion in [(Riehl and Shulman Reference Riehl and Shulman2017), A.1] on two ways to embed simplicial sets into simplicial spaces.
Remark 6.5. The interpretation of sHoTT can be done in greater generality. This is essentially a consequence of the main result in (Shulman Reference Shulman2019), where it is shown that any Grothendieck
$(\infty ,1)$
-topos gives a model for homotopy type theory. If
$\mathcal{C}$
is a type-theoretic model topos, then one can consider simplicial objects internal to
$\mathcal{E}$
. The resulting category of internal presheaves gives us again a model for sHoTT. For more details, we refer to [(Weinberger Reference Weinberger2022), Section 2].
Now, we have the semantical overview of sHoTT. This gives us the opportunity to say a few words on the necessity of contrasting the definitions, we make in the type theory with the existing ones in the bisimplicial sets model.
Remark 6.6. As we mentioned above, sHoTT can be interpreted in the category of bisimplicial sets. However, note that a type-theoretic statement translates internally to the bisimplicial sets model. A concrete example of this phenomenon is the characterization of Segal types. In general, each type is interpreted as a Reedy fibrant bisimplicial set. On the one hand, the equivalence
$ A^{\Delta ^2} \to A^{\Lambda _1^2}$
from Theorem 2.10 is interpreted as a trivial fibration in the Reedy model structure on
$ \mathbf{ssSet}$
. On the other hand, Segal spaces are defined by the Segal condition: a bisimplicial set
$A$
is a Segal space if, for all
$n\geq 2$
, the map
$A_n \to A_1 \times _{A_0} \cdots \times _{A_0} A_1$
is a trivial fibration in the Kan-Quillen model structure on
$ \mathbf{sSet}$
. [(Riehl and Shulman Reference Riehl and Shulman2017), A.3] shows that both conditions are equivalent.
Informally speaking, the meaning of an internal statement in the model could be stronger than its corresponding external analog. The results we present in the following sections show that this is not the case, that is the internal and external notions coincide.
6.1 Limits and colimits
We verify the consistency of our definitions of synthetic limits and colimits introduced in Definition 3.4. The notion of limits and colimits in Segal spaces we are considering are the ones presented in (Rasekh Reference Rasekh2023). The goal of this section is to prove the next result:
Theorem 6.7. The definitions of limits and colimits from Definition 3.4, under interpretation in the category of bisimplicial sets coincide with the definitions of limits and colimits from (Rasekh Reference Rasekh2023).
First, let us recall some standard notation and terminology we will use:
-
•
$\Delta$
will denote the category whose objects are the non-empty linearly ordered finite sets,
$[n] :\equiv \{0\leq 1 \leq \cdots \leq n \}$
with
$n\in \mathbb{N}$
, and morphisms the order-preserving maps between linearly ordered sets. -
• The representable presheaf
$\Delta ^{op} \to \mathbf{Set}$
given by
$[n] \in \Delta$
is noted by
$\Delta [n]$
. -
• For each
$ n\in \mathbb{N}$
,
$ F(n): \Delta ^{op} \times \Delta ^{op} \to \mathbf{Set}$
is defined asThis just means we are looking at the constant bisimplicial set
\begin{equation*} F(n)_{k,l} :\equiv \Delta ([k],[n]).\end{equation*}
$\Delta ([k],[n])$
. Two particular examples are
$F(1)$
and
$F(0)$
, the last of which is the terminal object.
-
• The category of simplicial sets is cartesian closed: for
$X,Y \in \mathbf{sSet}$
, the mapping simplicial set is defined as
\begin{equation*}Map(X,Y)_n :\equiv \mathbf{sSet}(X\times \Delta [n],Y).\end{equation*}
-
• The category of bisimplicial sets is cartesian closed: for
$X,Y \in \mathbf{ssSet}$
, the mapping bisimplicial set is defined as
\begin{equation*}(Y^X)_{kl} :\equiv \mathbf{ssSet}(F(k)\times \Delta [l] \times X,Y).\end{equation*}
-
• Decorated arrows “
$\twoheadrightarrow$
” indicate fibrations in the model structure. We will use this notation for the different model structures, but there is no room for confusion as the type of fibration is clear from the context.
Using Theorem 6.2, it is possible to conclude that simplicial type theory can be interpreted in bisimplicial sets. In particular,
$\mathsf {hom}$
types do coincide with
$hom$
spaces. From (Rezk Reference Rezk2001); if
$T$
is a Segal space, then, the mapping space between two objects
$a,b\in T_0$
is constructed via the pullback square of spaces:
On the other hand, from Theorem 2.3, we get
\begin{equation*}\Delta ^1\to A\simeq \sum \limits _{k:\partial \Delta ^1\to A}\left\langle \Delta ^1\to A|_k^{\partial \Delta ^1}\right\rangle ,\end{equation*}
and it is immediate that the type of functions
$\partial \Delta ^1\to A$
is equivalent to
$A\times A$
. Therefore, if
$A$
is a Segal type, then
$A^{\Delta ^1}$
is the total space of a family over
$A\times A$
whose fibers are
$\mathsf {hom}$
types. This family is exactly
${\mathsf {hom}} (-,-):A\times A\to \mathcal{U}$
. Under the standard interpretation,
$\Delta ^1$
is
$F(1)$
and
$\textbf {1}$
is
$F(0)$
. Given
$a,b:A$
we obtain
${\mathsf {hom}}_A(a,b)$
as the substitution along
$(a,b):{\textbf {1}}\to A\times A$
into
${\mathsf {hom}}_A(-,-)$
. When we interpret our type theory into simplicial spaces, this gives us the pullback square of bisimplicial sets,
We obtain the following comparison result:
Proposition 6.8.
Let
$ A$
be a Segal type and
$ a,b:A$
. The interpretation of
${\mathsf {hom}}_A(a,b)$
into bisimplicial sets is a constant simplicial space with value the Kan complex
$ map_A(a,b)$
.
Proof.
Note that
$ hom_A(a,b)_0=map_A(a,b)$
. Furthermore, from [(Riehl and Shulman Reference Riehl and Shulman2017), Proposition 8.29]
$ {\mathsf {hom}}_A (-,-): A \to A \to \mathcal{U}$
is a two-sided discrete fibration, so it is
$ A^{F(1)} \to A \times A$
. Hence, since each fiber
$ {\mathsf {hom}}_A(a,b)$
is discrete, it follows from Remark 6.4 that
$ hom_A(a,b)$
is a constant simplicial space at
$ map_A(a,b)$
.
The definition of a colimit from (Rasekh Reference Rasekh2023) is formulated as follows: let
$f:K\to A$
be a map of bisimplicial sets where
$A$
is a Segal space. The space of cocones under
$K$
is the Segal space
\begin{equation*} A_{f/}:\equiv F(0)\times _{A^K}(A^{K})^{F(1)}\times _{A^K}A. \end{equation*}
If
$a\in A$
, then, we can construct the space of objects under
$a$
by taking the map
$a:F(0)\to A$
and applying the previous definition.
The map
$f:K\to A$
has a colimit if the Segal space
$A_{f/}$
has an initial object. This is to say that there exists an object
$\sigma \in A_{f/}$
such that the map
$(A_{f/})_{\sigma /} \to A_{f/}$
is a trivial Reedy fibration. Let
$a\in A$
be an initial object. Therefore, for any object
$x\in A$
, we have the pullback:
The assumption that the map on the right is also an equivalence implies that
$hom_A(a,x)$
is equivalent to
$F(0)$
. This is to say that
$hom_A(a,x)$
is contractible, and it means that up to homotopy there is a unique map from
$a$
to
$x$
. Let us see first why this concept of initiality matches with Definition 2.26. In Proposition 6.8, we showed that
$hom_A(a,b)$
is a bisimplicial set constant at
$map_A(a,b)$
. It remains to show that equivalences are translated correctly. This follows immediately from [(Shulman Reference Shulman2015), Lemma 4.3]:
Lemma 6.9.
For a map
$f$
between fibrations
$p_1:E_1\to B$
and
$p_2:E_2\to B$
, the following are equivalent:
-
1.
$f$
is a weak equivalence.
-
2.
$\mathsf {IsEquiv}_B(f)\to B$
has a section.
-
3. There is a map
$ {\textbf {1}}\to \prod _B \mathsf {IsEquiv}_B(f)$
. -
4.
$\mathsf {IsEquiv}_B(f)\to B$
is an acyclic fibration.
Theorem 4.4.3 and Theorem 4.2.6 of (The Univalent Foundations Program 2013) imply that a function being an equivalence is equivalent to having contractible fibers. The object
$\mathsf {IsEquiv}_B(f)$
encodes this last fact, as is shown throughout [(Shulman Reference Shulman2015), Section 4]. Therefore, if a type is contractible, then under the standard interpretation, it is also contractible in the model. We can conclude that our notion Definition 2.26 of initiality coincides with the one in the semantics. All there is left to do is to compare the cocones with Definition 3.2. It is useful to see that the space
$A_{f/}$
is constructed with the successive pullbacks
Similar, for
$\mathsf {hom}$
types we find the following:
$(A^K)^{\Delta ^1}$
is the total space of the type family
${\mathsf {hom}}_{A^K}(-,-)$
over
$A^K\times A^K$
. Then, if we take
$f:K\to A$
and any
$x:A$
the type
${\mathsf {hom}}_{A^K}(f,\triangle x)$
is the substitution of
${\mathsf {hom}}_{A^K}(-,-)$
along
$(f,\triangle ):A \to A^K\times A^K$
. Thus, this is the pullback in the semantics:
Acknowledgments
This work is part of the author’s ongoing PhD thesis under the direction of Simon Henry. The author would like to thank his supervisor for his insights, comments, and suggestions, which greatly improved early versions of this paper. Thanks to the anonymous reviewers for their thorough reading of this work. The author also acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), under the grant reference number RGPIN-2020-06779, awarded to Simon Henry. The author is grateful for the support granted by the Department of Mathematics and Statistics of the University of Ottawa.
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