Animae as spaces

This chapter relates the abstract notion of an anima to classical homotopy theory. Recall from Chapter 1 that an anima is the homotopical analogue of a set, formally defined as an \(\infty \)-category in which all morphisms are invertible. The homotopy hypothesis makes the relationship precise: animae may be thought of as spaces up to (weak) homotopy equivalence. In particular, we will see that the classical constructions of cones, suspensions, loop spaces, homotopy pushouts, homotopy pullbacks and wedges have analogues phrased purely in terms of the \(\infty \)-category \(\An \).

The connection between spaces and animae is the content of Grothendieck’s homotopy hypothesis. To make it precise, we first need a way to pass from spaces to animae, that is, a functor \(\Top \to \An \). A topological space already behaves informally like an \(\infty \)-groupoid: its points are objects, paths are morphisms, homotopies between paths are higher morphisms, and this hierarchy continues indefinitely. This structure on \(X\) is neatly encoded by its singular complex: Recall from Example 1.6.8 that a topological space \(X\) has a simplicial set \[ \Sing (X)_n \; := \; \Hom _{\Top }(\abs {\Delta ^n},X). \] Thus its \(0\)-simplices are the points of \(X\), its \(1\)-simplices are paths, and its higher simplices encode homotopies and their higher coherences.

We may regard \(\Sing (X)\) as a simplicial anima via the inclusion \(\Set \hookrightarrow \An \) from Axiom L. We then define the underlying anima, or fundamental \(\infty \)-groupoid, of \(X\) by \[ \Piinfty {X} \quad := \quad \colim _{[n] \in \simp \catop } \Sing (X)_n \qin \An . \] This colimit glues together the singular simplices of \(X\). Although each individual topological simplex is contractible, the way these simplices meet along their faces retains the homotopy type of \(X\). Here it is crucial that the colimit is taken in \(\An \): it remembers the gluing homotopy coherently, rather than merely identifying points as an ordinary colimit in \(\Set \) would do. The construction is functorial in \(X\), and thus defines a functor \(\Pi _{\infty }\colon \Top \to \An \).

How faithfully does \(\Pi _{\infty }\) reflect the homotopy theory of spaces? As a first hint, one can show with a little effort that it sends every homotopy equivalence of topological spaces to an isomorphism of animae; in short, this holds because it preserves finite products and sends the interval to a contractible anima. In particular, the underlying anima of a topological space is a homotopy invariant of the space.

With some more effort, one can show that \(\Piinfty {X}\) is in fact a weak homotopy invariant of the space \(X\). Recall that algebraic topology studies spaces by assigning algebraic invariants to them. Among these invariants are the homotopy groups: generalizing the definition of the fundamental group as homotopy classes of loops in a space, one defines for each \(n \geq 1\) the \(n\)-th homotopy group of a space \(X\) at a basepoint \(x\) as the set \[ \pi _n(X,x) \quad := \quad [S^n,X]_* \] of pointed homotopy classes of pointed continuous maps from the topological \(n\)-sphere \(S^n \subseteq \R ^{n+1}\) into \(X\), with group structure defined via some form of concatenation. A continuous map \(f\colon X \to Y\) is called a weak homotopy equivalence if it induces a bijection on sets of path components and isomorphisms \(\pi _n(X,x) \xrightarrow {\cong } \pi _n(Y,f(x))\) on homotopy groups for all \(n \geq 1\) and all choices of basepoint \(x \in X\). An algebraic invariant of topological spaces is called a weak homotopy invariant if it sends every weak homotopy equivalence to an isomorphism in its target category; examples are the singular homology and cohomology of a space. A large part of algebraic topology is devoted to studying spaces up to weak homotopy equivalence.

If we allow our invariants of spaces to take values in arbitrary \(\infty \)-categories, then the functor \(\Pi _{\infty } \colon \Top \to \An \) constructed before turns out to be a weak homotopy invariant. Proving this requires classical results in simplicial homotopy theory that go beyond the scope of this book, and we refer to [Lurie (2026), Tag 00U2 and Tag 013P] for an exposition of this theory. The key fact relevant here is that a continuous map \(X \to Y\) of topological spaces is a weak homotopy equivalence if and only if the induced map of Kan complexes \(\Sing (X) \to \Sing (Y)\) is a simplicial homotopy equivalence. Since the geometric realization \(\abs {\Delta ^1}\) is contractible, it follows that \(\abs {-}\colon \sSet \to \An \) inverts simplicial homotopy equivalences, and we conclude that \(\Pi _{\infty }\) is a weak homotopy invariant.

Grothendieck’s homotopy hypothesis says that \(\Pi _{\infty }\) is not just any weak homotopy invariant; it is, in the strongest possible sense, the universal one. We enforce this behavior in our synthetic setup via the following axiom:

In light of the homotopy hypothesis, we may think of \(\An \) as the universal home for studying topological spaces up to weak homotopy equivalence. The rest of the chapter adds substance to this claim by transporting familiar constructions from algebraic topology into the world of animae. The main focus will be the classical theory of homotopy pushouts and pullbacks, which we recall in Section 2.1. In Section 2.2 we introduce a general framework for computing pushouts and pullbacks in localizations due to Cisinski (2019), Chapter 7, which we apply in Section 2.3 to show that \(\Pi _{\infty }\) sends homotopy pushouts and pullbacks of spaces to \(\infty \)-categorical pushouts and pullbacks of animae. This yields the basic constructions on pointed animae, such as fibers, cofibers, suspensions, and loop spaces, developed in Section 2.4, which will become crucial in the remainder of the book.

Generated from the authoritative LaTeX source.