Real-cohesive homotopy type theory

Two new papers have recently appeared online:

Both of them have fairly chatty introductions, so I’ll try to restrain myself from pontificating at length here about their contents. Just go read the introductions. Instead I’ll say a few words about how these papers came about and how they are related to each other.

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Posted in Applications, Foundations, Paper | 12 Comments

A new class of models for the univalence axiom

First of all, in case anyone missed it, Chris Kapulkin recently wrote a guest post at the n-category cafe summarizing the current state of the art regarding “homotopy type theory as the internal language of higher categories”.

I’ve just posted a preprint which improves that state a bit, providing a version of “Lang(C)” containing univalent strict universes for a wider class of (∞,1)-toposes C:

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Posted in Models, Paper, Univalence | 4 Comments

Constructing the Propositional Truncation using Nonrecursive HITs

In this post, I want to talk about a construction of the propositional truncation of an arbitrary type using only non-recursive HITs. The whole construction is formalized in the new proof assistant Lean, and available on Github. I’ll write another blog post explaining more about Lean in August.

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Posted in Code, Higher Inductive Types | 25 Comments

Universal Properties of Truncations

Some days ago at the HoTT/UF workshop in Warsaw (which was a great event!), I have talked about functions out of truncations. I have focussed on the propositional truncation \Vert - \Vert, and I want to write this blog post in case someone could not attend the talk but is interested nevertheless. There are also my slides and the arXiv paper.
The topic of my talk can be summarised as

Question: What is the type \Vert A \Vert \to B ?

This question is very easy if B is propositional, because then we have

Partial answer (1): If B is propositional, then (\Vert A \Vert \to B) \simeq (A \to B)

by the usual universal property, with the equivalence given by the canonical map. Without an assumption on B, it is much harder.

The rest of this post contains a special case which already contains some of the important ideas. It is a bit lengthy, and in case you don’t have time to read everything, here is the

Main result (general universal property of the propositional truncation).
In a dependent type theory with at least \mathbf{1}, \Sigma-, \Pi-, and identity types, which furthermore has Reedy \omega^{\mathsf{op}}-limits (“infinite \Sigma-types”), we can define the type of coherently constant functions A \xrightarrow {\omega} B as the type of natural transformations between type-valued presheaves. If the type theory has propositional truncations, we can construct a canonical map from \Vert A \Vert \to B to A \xrightarrow {\omega} B. If the type theory further has function extensionality, then this canonical map is an equivalence.
If B is known to be n-truncated for some fixed n, we can drop the assumption of Reedy \omega^{\mathsf{op}}-limits and perform the whole construction in “standard syntactical” HoTT. This describes how functions \Vert A \Vert \to B can be defined if B is not known to be propositional, and it streamlines the usual approach of finding a propositional Q with A \to Q and Q \to B.


Here comes the long version of this blog post. So, what is a function g : \Vert A \Vert \to B? If we think of elements of \Vert A \Vert as anonymous inhabitants of A, we could expect that such a g is “the same” as a function f : A \to B which “cannot look at its input”. But then, how can we specify what it means to “not look at its input” internally? A first attempt could be requesting that f is weakly constant, \mathsf{wconst}_f :\equiv \Pi_{x,y:A} f(x) = f(y). Indeed, it has been shown:

Partial answer (2): If B is a set (h-set, 0-truncated), then (\Vert A \Vert \to B) \simeq (\Sigma (f : A \to B). \mathsf{wconst}_f), where the function from left to right is the canonical one.

It is not surprising that we still need this strong condition on B if we want weak constancy to be a sufficient answer: just throwing in a bunch of paths, which might or might not “fit together” (we just don’t know anything) seems wrong. Indeed, from Mike’s recent construction, we know that the statement does become false (contradicts univalence) without this requirement.

Given a function f : A \to B and a proof c : \mathsf{wconst}_f, we can try to fix the problem that the paths given by c “do not fit together” by throwing in a coherence proof, i.e. an element of \mathsf{coh}_{f,c} :\equiv \Pi_{x,y,z:A} c(x,y) \cdot c(y,z) = c(x,z). We should already know that this will introduce its own problems and thus not fix everything, but at least, we get:

Partial answer (3): If B is 1-truncated, then (\Vert A \Vert \to B) \simeq (\Sigma (f : A \to B). \Sigma (c:\mathsf{wconst}_f). \mathsf{coh}_{f,c}), again given by a canonical map from left to right.

In my talk, I have given a proof of “partial answer (3)” via “reasoning with equivalences” (slides p. 5 ff). I start by assuming a point a_0 : A. The type B is then equivalent to the following (explanation below):

\Sigma (f_1 : B).
\Sigma (f : A \to B). \Sigma (c_1 : \Pi_{a:A} f(a) = f_1).
\Sigma (c : \mathsf{const}_f). \Sigma(d_1 : \Pi_{a^1, a^2:A} c(a^1,a^2) \cdot c_1(a^2) = c_1(a^1)
\Sigma (c_2 : f(a_0) = f_1). \Sigma (d_3 : c(a_0,a_0) \cdot c_1(a_0) = c_2).
\Sigma (d : \mathsf{coh}_{f,c_0}).
\Sigma (d_2 : \Pi_{a:A} c(a_0,a) \cdot c_1(a) = c_2).
\mathbf{1}

In the above long nested \Sigma-type, the first line is just the B that we start with. The second line adds two factors/\Sigma-components which, by function extensionality, cancel each other out (they form a singleton). The same is true for the pair in the third line, and for the pair in the fourth line. Lines five and six look different, but they are not really; it’s just that their “partners” (which would complete them to singletons) are hard to write down and, at the same time, contractible; thus, they are omitted. As B is assumed to be a 1-type, it is easy to see that the \Sigma-components in lines five and six are both propositional, and it’s also easy to see that the other \Sigma-components imply that they are both inhabited. Of course, the seventh line does nothing.

Simply by re-ordering the \Sigma-components in the above type, and not doing anything else, we get:

\Sigma (f : A \to B). \Sigma(c : \mathsf{const}_f). \Sigma(d : \mathsf{coh}_{f,c}).
\Sigma (f_1 : B) \Sigma(c_2 : f(a_0) = f_1).
\Sigma (c_1 : \Pi_{a:A} f(a) = f_1). \Sigma(d_2 : \Pi_{a:A} c(a_0,a) \cdot c_1(a) = c_2).
\Sigma (d_1 : \Pi_{a^1, a^2:A} c(a^1,a^2) \cdot c_1(a^2) = c_1(a^1)).
\Sigma (d_3 : c(a_0,a_0) \cdot c_1(a_0) = c_2).
\mathbf{1}

By the same argument as before, the components in the pair in line two cancel each other out (i.e. the pair is contractible). The same is true for line three. Line four and five are propositional as B is 1-truncated, but easily seen to be inhabited. Thus, the whole nested \Sigma-type is equivalent to \Sigma (f : A \to B). \Sigma(c : \mathsf{const}_f). \mathsf{coh}_{f,c}.
In summary, we have constructed an equivalence B \simeq \Sigma (f : A \to B). \Sigma(c : \mathsf{const}_f). \mathsf{coh}_{f,c}. By going through the construction step-by-step, we see that the function part of the equivalence (map from left to right) is the canonical one (let’s write \mathsf{canon}_1), mapping b : B to the triple (\lambda \_. b, \lambda \_ \_ . \mathsf{refl}_b , \lambda \_ \_ \_ . \mathsf{refl}_{\mathsf{refl}_b}). Before we have started the construction, we have assumed a point a_0 : A. But, and this is the crucial observation, the function \mathsf{canon}_1 does not depend on a_0! So, if the assumption A implies that \mathsf{canon}_1 is an equivalence, then \Vert A \Vert is already enough. Thus, we have \Vert A \Vert \to \big(B \simeq \Sigma (f : A \to B). \Sigma(c : \mathsf{const}_f). \mathsf{coh}_{f,c}\big). We can move the \Vert A \Vert \to-part to both sides of the equivalence, and on the right-hand side, we can apply the usual “distributivity of \Sigma and \Pi (or \to)”, to move the \Vert A \Vert \to into each \Sigma-component; but in each \Sigma-component, the \Vert A \Vert gets eaten by an A (we have that \Vert A \Vert \to \Pi_{a : A}\ldots and \Pi_{a:A} \ldots are equivalent), which gives us the claimed result.

The strategy we have used is very minimalistic. It does not need any “technology”: we just add and remove contractible pairs. For this “expanding and contracting” strategy, we have really only used very basic components of type theory (\Sigma, \Pi, identity types with function extensionality, propositional truncation, but even that only in the very end). If we want to weaken the requirement on B by one more level (i.e. if we want to derive a “universal property” which characterises \Vert A \Vert \to B if B is 2-truncated), we have to add one more coherence condition (which ensures that the \mathsf{coh}_{f,c}-component is well-behaved). The core idea is that this expanding and contracting strategy can be done for any truncation level of B, even if B is not known to be n-truncated at all, where the tower of conditions becomes infinite. I call an element of this “infinite \Sigma-type” a coherently constant function from A to B.

The idea is that the \Sigma-components (f : A \to B) and (c : \mathsf{const}_f) and (d : \mathsf{coh}_{f,c}) we used in the special case can be seen as components of a natural transformation between [2]-truncated semi-simplicial types. By semi-simplicial type, I mean a Reedy fibrant diagram \Delta_+^{\mathsf{op}} \to \mathcal{C}. By [2]-truncated semi-simplicial types, I mean the “initial segments” of semi-simplicial types, the first three components X_0, X_1, X_2 as described here.
The first diagram we need is what I call the “trivial semi-simplicial type” over A, written T \! A, and its initial part is given by T \! A_{[0]} :\equiv A, T \! A_{[1]} :\equiv A \times A, and T \! A_{[2]} :\equiv A \times A \times A. If we use the “fibred” notation (i.e. give the fibres over the matching objects), this would be T \! A_{[0]} :\equiv A, T \! A_{[1]}(a_1,a_2) :\equiv \mathbf{1}, T \! A_{[2]}(a_1,a_2,a_3,\_,\_,\_) :\equiv \mathbf{1}. In the terminology of simplicial sets, this is the [0]-coskeleton of [the diagram that is constantly] A.
The second important diagram, I call it the equality semi-simplicial type over B, is written E \! B. One potential definition for the lowest levels would be given by (I only give the fibres this time): E \! B_{[0]} :\equiv B, E \! B_{[1]}(b_1,b_2) :\equiv b_1 = b_2, E \! B_{[2]}(b_1,b_2,b_3,p_{12},p_{23},p_{13}) :\equiv p_{12} \cdot p_{23} = p_{13}. This is a fibrant replacement of B. (We are lucky because T \! A is already fibrant; otherwise, we should have constructed a fibrant replacement as well.)
If we now check what a (strict) natural transformation between T \! A and E \! B (viewed as diagrams over the full subcategory of \Delta_+^{\mathsf{op}} with objects [0],[1],[2]) is, it is easy to see that the [0]-component is exactly a map f : A \to B, the [1]-component is exactly a proof c : \mathsf{const}_f, and the [2]-component is just a proof d : \mathsf{coh}_{f,c}. (The type of such natural transformations is given by the limit of the exponential of T \! A and E \! B.)

However, even with this idea, and with the above proof for the case that B is 1-truncated, generalising this proof to the infinite case with infinitely many coherence conditions requires some ideas and a couple of technical steps which, for me, have been quite difficult. Therefore, it has taken me a very long time to write this up cleanly in an article. I am very grateful to the reviewers (this work is going to appear in the post-proceedings of TYPES’14) and to Steve (my thesis examiner) for many suggestions and interesting connections they have pointed out. In particular, one reviewer has remarked that the main result (the equivalence of coherently constant functions A \xrightarrow{\omega} B and maps out of the truncation \Vert A \Vert \to B) is a type-theoretic version of Proposition 6.2.3.4 in Lurie’s Higher Topos Theory; and Vladimir has pointed out the connection to the work of his former student Alexander Vishik. Unfortunately, both are among the many things that I have yet to understand, but there is certainly a lot to explore. If you can see further connections which I have not mentioned here or in the paper, then this is very likely because I am not aware of them, and I’d be happy to hear about it!

Posted in Foundations, Homotopy Theory, Models, Paper, Talk | 3 Comments

Modules for Modalities

As defined in chapter 7 of the book, a modality is an operation on types that behaves somewhat like the n-truncation. Specifically, it consists of a collection of types, called the modal ones, together with a way to turn any type A into a modal one \bigcirc A, with a universal property making the modal types a reflective subcategory (or more precisely, sub-(∞,1)-category) of the category of all types. Moreover, the modal types are assumed to be closed under Σs (closure under some other type formers like Π is automatic).

We called them “modalities” because under propositions-as-(some)-types, they look like the classical notion of a modal operator in logic: a unary operation on propositions. Since these are “monadic” modalities — in particular, we have A \to \bigcirc A rather than the other way around — they are most closely analogous to the “possibility” modality of classical modal logic. But since they act on all types, not just mere-propositions, for emphasis we might call them higher modalities.

The example of n-truncation shows that there are interesting new modalities in a homotopy world; some other examples are mentioned in the exercises of chapter 7. Moreover, most of the basic theory of n-truncation — essentially, any aspect of it that involves only one value of n — is actually true for any modality.

Over the last year, I’ve implemented this basic theory of modalities in the HoTT Coq library. In the process, I found one minor error in the book, and learned a lot about modalities and about Coq. For instance, my post about universal properties grew out of looking for the best way to define modalities.

In this post, I want to talk about something else I learned about while formalizing modalities: Coq’s modules, and in particular their universe-polymorphic nature. (This post is somewhat overdue in that everything I’ll be talking about has been implemented for several months; I just haven’t had time to blog about it until now.)

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Posted in Code, Programming | 19 Comments

Not every weakly constant function is conditionally constant

As discussed at length on the mailing list some time ago, there are several different things that one might mean by saying that a function f:A\to B is “constant”. Here is my preferred terminology:

  • f is constant if we have b:B such that f(a)=b for all a:A.
    This is equivalent to saying that f factors through \mathbf{1}.
  • f is conditionally constant if it factors through \Vert A \Vert.
  • f is weakly constant if for all a_1,a_2:A we have f(a_1)=f(a_2).

In particular, the identity function of \emptyset is conditionally constant, but not constant. I don’t have a problem with that; getting definitions right often means that they behave slightly oddly on the empty set (until we get used to it). The term “weakly constant” was introduced by Kraus, Escardo, Coquand, and Altenkirch, although they immediately dropped the adjective for the rest of their paper, which I will not do. The term “conditionally constant” is intended to indicate that f is, or would be, constant, as soon as its domain is inhabited.

It’s obvious that every constant function is conditionally constant, and \mathrm{id}_{\emptyset} shows the converse fails. Similarly, it’s easy to show that conditionally constant implies weakly constant. KECA showed that the converse holds if either B is a set or \Vert A \Vert \to A, and conjectured that it fails in general. In this post I will describe a counterexample proving this conjecture.

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Posted in Univalence | 10 Comments

Double Groupoids and Crossed Modules in HoTT

The past eight months I spent at CMU for my master thesis project. I ended up formalizing some algebraic structures used in Ronald Brown’s book “Non-Abelian Algebraic Topology”: Double groupoids with thin structure and crossed modules over groupoids.

As the language for the formalizations I chose the newly developed theorem prover Lean. Lean is developed by Leonardo de Moura at Microsoft Research and provides a dependently typed language similar to Agda or Coq. Currently, there are two modes for Lean: A proof-irrelevant mode and a HoTT mode. The HoTT library was written at the same time as my project, mainly by Floris van Doorn and, for a smaller part, me. You can find papers about Lean and its elaboration algorithm on the website, as well as a link to a version of Lean that runs in your web browser.

Double groupoids are a special case double categories which, besides objects and morphisms, specify a set of two-cells for each (not necessarily commuting) square diagram in a way such that the two-cells form a category with respect to vertical and to horizontal composition. That includes the existence of vertically and horizontally degenerate fillers of square diagrams with the identity on two opposing sides and the same morphism on the other side. The vertically degenerate squares should distribute over the horizontal composition of morphisms and vice versa. Furthermore, the vertically and horizontally degenerate square along the identity should be the same. A last axiom is the interchange law which ensures that, in the case of two-cells which are composable in a 2×2-grid, it doesn’t matter whether to first compose vertically or horizontally.

I decided to formalize double categories in the same way categories are usually formalized using dependent types using a type of two-cells depending on four points and four morphisms:

D_2 : \prod_{a, b, c, d : D_0} \text{hom}(a,b) \to \text{hom}(c,d) \to \text{hom}(a,c) \to \text{hom}(b,d) \to \mathcal{U}.In the formalization, I first define a worm precategory which only allows for two-cell composition in one direction. Then, I use Lean’s inheritance mechanism to get the full notion of a double (pre)category:


structure worm_precat {D₀ : Type} (C  : precategory D₀)
  (D₂ : Π ⦃a b c d : D₀⦄
    (f : hom a b) (g : hom c d) (h : hom a c) (i : hom b d), Type) :=
  (comp₁ : proof Π ⦃a b c₁ d₁ c₂ d₂ : D₀⦄
    ⦃f₁ : hom a b⦄ ⦃g₁ : hom c₁ d₁⦄ ⦃h₁ : hom a c₁⦄ ⦃i₁ : hom b d₁⦄
    ⦃g₂ : hom c₂ d₂⦄ ⦃h₂ : hom c₁ c₂⦄ ⦃i₂ : hom d₁ d₂⦄,
    (D₂ g₁ g₂ h₂ i₂) → (D₂ f₁ g₁ h₁ i₁)
    → (@D₂ a b c₂ d₂ f₁ g₂ (h₂ ∘ h₁) (i₂ ∘ i₁)) qed)
  (ID₁ : proof Π ⦃a b : D₀⦄ (f : hom a b), D₂ f f (ID a) (ID b) qed)
  (assoc₁ : proof Π ⦃a b c₁ d₁ c₂ d₂ c₃ d₃ : D₀⦄
    ⦃f  : hom a b⦄   ⦃g₁ : hom c₁ d₁⦄ ⦃h₁ : hom a c₁⦄ ⦃i₁ : hom b d₁⦄
    ⦃g₂ : hom c₂ d₂⦄ ⦃h₂ : hom c₁ c₂⦄ ⦃i₂ : hom d₁ d₂⦄
    ⦃g₃ : hom c₃ d₃⦄ ⦃h₃ : hom c₂ c₃⦄ ⦃i₃ : hom d₂ d₃⦄
    (w : D₂ g₂ g₃ h₃ i₃) (v : D₂ g₁ g₂ h₂ i₂) (u : D₂ f g₁ h₁ i₁),
    (assoc i₃ i₂ i₁) ▹ ((assoc h₃ h₂ h₁) ▹
        (comp₁ w (comp₁ v u))) = (comp₁ (comp₁ w v) u) qed)
  ...

structure dbl_precat {D₀ : Type} (C : precategory D₀)
  (D₂ : Π ⦃a b c d : D₀⦄
    (f : hom a b) (g : hom c d) (h : hom a c) (i : hom b d), Type)
  extends worm_precat C D₂,
    worm_precat C (λ ⦃a b c d : D₀⦄ f g h i, D₂ h i f g)
  renaming comp₁→comp₂ ID₁→ID₂ assoc₁→assoc₂
    id_left₁→id_left₂ id_right₁→id_right₂ homH'→homH'_dontuse :=
  ...
  (interchange : proof Π {a₀₀ a₀₁ a₀₂ a₁₀ a₁₁ a₁₂ a₂₀ a₂₁ a₂₂ : D₀}
    {f₀₀ : hom a₀₀ a₀₁} {f₀₁ : hom a₀₁ a₀₂} {f₁₀ : hom a₁₀ a₁₁}
    {f₁₁ : hom a₁₁ a₁₂} {f₂₀ : hom a₂₀ a₂₁} {f₂₁ : hom a₂₁ a₂₂}
    {g₀₀ : hom a₀₀ a₁₀} {g₀₁ : hom a₀₁ a₁₁} {g₀₂ : hom a₀₂ a₁₂}
    {g₁₀ : hom a₁₀ a₂₀} {g₁₁ : hom a₁₁ a₂₁} {g₁₂ : hom a₁₂ a₂₂}
    (x : D₂ f₁₁ f₂₁ g₁₁ g₁₂) (w : D₂ f₁₀ f₂₀ g₁₀ g₁₁)
    (v : D₂ f₀₁ f₁₁ g₀₁ g₀₂) (u : D₂ f₀₀ f₁₀ g₀₀ g₀₁),
    comp₁ (comp₂ x w) (comp₂ v u) = comp₂ (comp₁ x v) (comp₁ w u) qed)

Double groupoids are double categories where all three categories involved — the underlying category of points and morphisms, the vertical category of two-cells, and the horizontal category of two-cells — are groupoids and that are equipped with a thin structure: A selection of a unique thin filler of each commutative square shell which is closed under composition and degenerate squares.

It turns out that a category equivalent to the one of double groupoids is the category of crossed modules over a groupoid. A crossed module is defined as a groupoid $P$ together with a family (M_p)_{p \in P} of groups (or, equivalently, a totally disconnected groupoid) on the objects of P and for each p \in P a group homomorphism \mu_p : M_p \to \text{hom}(p,p) and a groupoid action \phi of P on the (M_p)_{p \in P}. The family of homomorphisms and the action should interact by resembling conjugation via \mu_q(\phi(f,x)) = f \circ \mu_p(x) \circ f^{-1} and \phi(\mu_p(c),x) = c \cdot x \cdot c^{-1} for each p, q \in P, $f \in \text{hom}_P(p,q)$ and c, x \in M_p.


structure xmod {P₀ : Type} [P : groupoid P₀] (M : P₀ → Group) :=
  (P₀_hset : is_hset P₀)
  (μ : Π ⦃p : P₀⦄, M p → hom p p)
  (μ_respect_comp : Π ⦃p : P₀⦄ (b a : M p), μ (b * a) = μ b ∘ μ a)
  (μ_respect_id : Π (p : P₀), μ 1 = ID p)
  (φ : Π ⦃p q : P₀⦄, hom p q → M p → M q)
  (φ_respect_id : Π ⦃p : P₀⦄ (x : M p), φ (ID p) x = x)
  (φ_respect_P_comp : Π ⦃p q r : P₀⦄ (b : hom q r) (a : hom p q) (x : M p),
    φ (b ∘ a) x = φ b (φ a x))
  (φ_respect_M_comp : Π ⦃p q : P₀⦄ (a : hom p q) (y x : M p),
    φ a (y * x) = (φ a y) * (φ a x))
  (CM1 : Π ⦃p q : P₀⦄ (a : hom p q) (x : M p), μ (φ a x) = a ∘ (μ x) ∘ a⁻¹)
  (CM2 : Π ⦃p : P₀⦄ (c x : M p), φ (μ c) x = c * (x * c⁻¹ᵍ))

After formlizing both categories, DGpd and Xmod including the definition of their respective morphisms, I defined the functors \gamma and \lambda which establish their equivalence.


  definition gamma.functor :
    functor Cat_dbl_gpd.{l₁ l₂ l₃} Cat_xmod.{(max l₁ l₂) l₂ l₃} :=
  begin
    fapply functor.mk,
      intro G, apply (gamma.on_objects G),
      intros [G, H, F], apply (gamma.on_morphisms F),
      intro G, cases G,
        fapply xmod_morphism_congr, apply idp, apply idp,
        repeat ( apply eq_of_homotopy ; intros), cases x_1, apply idp,
    ...
  end

Since establishing the functors was quite tedious, I didn’t finish proving the equivalence, but I might do that in the future.

As a little application on 2-types I instantiated the fundamental double groupoid of a 2-type X which is presented by a set C and a 1-type A by arbitrary functions \iota : C \to A and \iota' : A \to X. Here, the type of objects is the set C, the set of morphisms between a, b : C is the identity type \iota(a) =_A \iota(b), and the set of two-cells associated to four points a, b, c, d : C and paths f, g, h, i on the top, bottom, left and right of a diagram is the set \text{ap}_{\iota'}(h) \cdot \text{ap}_{\iota'}(g) = \text{ap}_{\iota'}(f) \cdot \text{ap}_{\iota'}(i). This could serve as the starting point to formalize a 2-dimensional Seifert-van Kampen theorem using double groupoids or crossed modules.

You can find my thesis here. The Lean code used can be found on github, due to recent changes in Lean, it most probably won’t compile with the newest version of Lean but with an older snapshot that can be found here.

Posted in Code, Homotopy Theory | 31 Comments