Equipping weak equivalences with algebraic structure


We investigate the extent to which the weak equivalences in a model category can be equipped with algebraic structure. We prove, for instance, that there exists a monad T such that a morphism of topological spaces admits T-algebra structure if and only it is a weak homotopy equivalence. Likewise for quasi-isomorphisms and many other examples. The basic trick is to consider injectivity in arrow categories. Using algebraic injectivity and cone injectivity we obtain general results about the extent to which the weak equivalences in a combinatorial model category can be equipped with algebraic structure.

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The author gratefully acknowledges the support of an Australian Research Council Discovery Grant DP160101519 and the support of the Grant Agency of the Czech Republic under the grant 19-00902S. Particular thanks are due to Emily Riehl whose interest in an algebraic version of Smith’s theorem got me thinking about this topic and to Lukáš Vokřínek who helped me to see the connection between \(Ex_{\infty }\) and the generating cones for simplicial sets. Thanks also to the organisers of the PSSL101 in Leeds for providing the opportunity to present this work, and to the members of the Australian Category Seminar for listening to me speak about it.

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Appendix A: Free algebras for pointed endofunctors

Appendix A: Free algebras for pointed endofunctors

In order to make the proof of Theorem 11 accessible to a broader audience we now describe in detail the construction of free algebras for pointed endofunctors. The classical reference is [20], specifically Theorems 14.3 and 15.6. Here we take a different approach to essentially the same result. Our approach is based upon, and is a straightforward modification of, Koubek and Reiterman’s elegant construction of the free algebra on an endofunctor [21]. One of the attractive features of this approach is that it emphasises the explicit formulae involved—see Proposition 21 below—by focusing not only on the free algebra but also on the free algebraic chain.

To begin with, a chain is a functor \(X:Ord\rightarrow {\mathcal {C}}\) on the posetal category of ordinals, whilst a chain map is a natural transformation. Given a pointed endofunctor \((T,\eta )\) on \({\mathcal {C}}\) an algebraic chain (Xx) is a chain X together with, for each ordinal n, a map \(x_n:TX_n \rightarrow X_{n+1}\) satisfying

  • for all n

  • and for all \(n < m\) the diagram



A morphism \(f:(X,x) \rightarrow (Y,y)\) of algebraic chains is a chain map that commutes with the \(x_n\) and \(y_n\) for all n. These are the morphisms of the category \(\mathrm{T\hbox {-}Alg} _\infty \) of algebraic chains.

Example 20

Let J be a set of morphisms in \({\mathcal {C}}\). In Sect. 2.3 we described the pointed endofunctor \((R,\eta )\) whose algebras are algebraic injectives. Using the construction of R in (2.5) we see that an algebraic chain is a chain X together with, for each lifting problem \((\alpha :A \rightarrow B \in J, f:A \rightarrow X_{n})\), a filler \(x_{n}(\alpha ,f)\) rendering the left square below commutative.

These fillers must satisfy the indicated compatibility for \(n<m\).

There is a forgetful functor \(V:\mathrm{T\hbox {-}Alg} _\infty \rightarrow {\mathcal {C}}\) sending (Xx) to \(X_0\). Our first goal is to show that if \({\mathcal {C}}\) is cocomplete then V has a left adjoint.

To this end we first observe that Eq. (A.2) holds for all \(n < m\) if it does so in the cases (a) \(m=n+1\) and (b) m is a limit ordinal. Now consider a chain X equipped with maps \(x_{n}:TX_n \rightarrow X_{n+1}\) satisfying (A.1). Then case (a) of (A.2) becomes the assertion that for all n the diagram


is a fork. Case (b) of (A.2) asserts that for all limit ordinals m and \(n<m\) the diagram

is a fork. To see this, use that \(x_{m} \circ \eta _{X_{m}} = j_{m}^{m+1}\). In the presence of filtered colimits this equally asserts that for each limit ordinal m the diagram


is a fork.

Proposition 21

If \({\mathcal {C}}\) is cocomplete then V has a left adjoint whose value at \(X \in {\mathcal {C}}\) is the algebraic chain \(X_{\bullet }\) with values:

  • \(X_{0}=X\), \(X_{1}=TX\), \(j_{0}^{1}=\eta _{X}:X \rightarrow TX\) and \(x_0=1:TX \rightarrow TX\).

  • At an ordinal of the form \(n+2\) the object \(X_{n+2}\) is the coequaliser

    with \(j_{n+1}^{n+2}=x_{n+1} \circ \eta _{X_{n+1}}\).

  • At a limit ordinal m,

    • \(X_{m} = col_{n < m} X_{n}\) with the connecting maps \(j_n^m\) the colimit inclusions.

    • \(X_{m + 1}\) is the coequaliser

      with \(j_{m}^{m+1}=x_{m} \circ \eta _{X_{m}}\).


The unit of the adjunction will be the identity—so, we are to show that given \(f:X \rightarrow Y_{0}=V(Y,y)\) there exists a unique map \(f:X_\bullet \rightarrow (Y,y)\) of algebraic chains with \(f_{0}=f\). The required commutativity below left

forces us to set \(f_{1}=y_{0} \circ Tf\). The map \(f_{n+2}\) must render the right square in the diagram above right commutative. But since the two back squares serially commute and the bottom row is a fork there exists a unique map from the coequaliser \(X_{n+2}\) rendering the right square commutative. This uniquely specifies \(f_n\) for \(n < \omega \). At a limit ordinal m, \(f_m:X_{m}=col_{n < m} X_{n} \rightarrow Y_{m}\) is the unique map from the colimit commuting with the connecting maps—which it must do to form a morphism of chains. At the successor of a limit ordinal m there is a unique map \(f_{m+1}:X_{m+1} \rightarrow Y_{m+1}\) from the coequaliser satisfying \(f_{m+1} \circ x_{m} = y_{m} \circ Tf_{m}\), as required. \(\square \)

The usual forgetful functor \(U:\mathrm{T\hbox {-}Alg} \rightarrow {\mathcal {C}}\) factors through \(V:\mathrm{T\hbox {-}Alg} _\infty \rightarrow {\mathcal {C}}\) via a functor \(\Delta :\mathrm{Alg} \rightarrow \mathrm{Alg} _\infty \): this sends (Xx) to the constant chain on X equipped with \(x_n=x\) for all n. A chain X is said to stabilise at an ordinal n if for all \(m>n\) the map \(j_{n,m}:X_n \rightarrow X_m\) is invertible. Observe that if an algebraic chain (Xx) stabilises at n then \(X_{n}\) equipped with the T-algebra structure

$$\begin{aligned} (j_n^{n+1})^{-1} \circ x_n:TX_n \rightarrow X_{n+1} \cong X_n \end{aligned}$$

is a reflection of (Xx) along \(\Delta \). In particular:

Proposition 22

If \(X_{\bullet }\) stabilises at n then \(X_{n}\), with structure map as in (A.5), is the free T-algebra on X.

Accordingly we examine circumstances under which each \(X_{\bullet }\) stabilises. In the following the term chain of length n refers to a functor \(X:Ord_{<n} \rightarrow {\mathcal {C}}\) from the full subcategory of ordinals less than n.

Proposition 23

If T preserves the colimit \(X_{m}=col_{n<m}X_{n}\) for m a limit ordinal then \(X_{\bullet }\) stabilises at the ordinal m.


Firstly one shows that \(j_m^{m+1}:X_{m} \rightarrow X_{m+1}\) is invertible. To see this observe that the morphisms \(x_{n}:TX_n \rightarrow X_{n+1}\) form a morphism of chains of length m, and so induce a map \(x_m:TX_m \rightarrow X_m\) between the colimits. This has the universal property of the coequaliser \(x_m:TX_{m} \rightarrow X_{m+1}\) whereby the comparison \(j_m^{m+1}\) between the two coequalisers is invertible.

Now the coequaliser formulae allow to us to prove that if for some k the map \(j_k^{k+1}\) is invertible then so is \(j_{k+1}^{k+2}\) and, likewise, that if \(j_k^l\) is invertible for \(k < l\) with l a limit ordinal then \(j_l^{l+1}\) is invertible. Given that \(j_m^{m+1}\) is invertible it easily follows from these facts, using transfinite induction, that each \(j_m^n\) is invertible for all \(n>m\). \(\square \)

Theorem 24

Let \((T,\eta )\) be a pointed endofunctor on a cocomplete category \({\mathcal {C}}\). If either

  1. (1)

    T preserves colimits of n-chains for some limit ordinal n, or

  2. (2)

    \({\mathcal {C}}\) is equipped with a well copowered proper factorisation system \((\mathcal {E},\mathcal {M})\) such that T preserves colimits of \(\mathcal {M} \)-chains of length n for some limit ordinal n.

Then free T-algebras exist: namely, each algebraic chain \(X_{\bullet }\) stabilises and its point of stabilisation, with algebra structure as in (A.5), is the free T-algebra on X.


Assuming (1) the conclusion holds on combining the three preceding propositions. Assuming (2), it suffices to show that if A is any chain, then there exists a limit ordinal m such that T preserves the colimit of the chain \((A_{n})_{n < m}\) of length n. This is the content of a clever lemma from Section 8.5 of Koubek and Reiterman [21]. See also Proposition 4.1 of [20] for a helpful proof of that result. \(\square \)

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Bourke, J. Equipping weak equivalences with algebraic structure. Math. Z. 294, 995–1019 (2020). https://doi.org/10.1007/s00209-019-02305-w

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  • Monads
  • Algebraic injectives
  • Weak equivalences

Mathematics Subject Classification

  • Primary 55U35
  • Secondary 18C35