nLab (∞,1)-algebraic theory



Higher algebra

(,1)(\infty,1)-Category theory



In as far as an algebraic theory or Lawvere theory is nothing but a small category with finite products and an algebra for the theory a product-preserving functor to Set, the notion has an evident generalization to higher category theory and in particular to (∞,1)-category theory.



An (,1)(\infty,1)-Lawvere theory is (given by a syntactic (,1)(\infty,1)-category that is) an (∞,1)-category CC with finite (∞,1)-products. An (,1)(\infty,1)-algebra for the theory is an (∞,1)-functor CC \to ∞Grpd that preserves these products.

The (,1)(\infty,1)-category of ∞-algebras over an (∞,1)-algebraic theory is the full sub-(∞,1)-category

Alg (,1)(C)PSh (,1)(C op) Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})

of the (∞,1)-category of (∞,1)-presheaves on C opC^{op} on the product-preserving (,1)(\infty,1)-functors

In a full (,1)(\infty,1)-category theoretic context this appears as HTT, def. A definition in terms of simplicially enriched categories and the model structure on sSet-categories to present (,1)(\infty,1)-categories is in Ros. The introduction of that article lists further and older occurences of this definition.



Let CC be an (∞,1)-category with finite products. Then

This is HTT, prop.


There are various model category presentations of Alg (,1)(C)PSh (,1)(C op)Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op}).

Recall that the (∞,1)-category of (∞,1)-presheaves PSh (,1)(C op)PSh_{(\infty,1)}(C^{op}) itself is modeled by the model structure on simplicial presheaves

PSh (,1)(C op)[T,sSet] , PSh_{(\infty,1)}(C^{op}) \simeq [T, sSet]^\circ \,,

where we regard TT as a Kan complex-enriched category and have on the right the sSet-enriched functor category with the projective or injective model structure, and () (-)^\circ denoting the full enriched subcategory on fibrant-cofibrant objects.

This says in particular that every weak (,1)(\infty,1)-functor f:TGrpf : T \to \infty \mathrm{Grp} is equivalent to a rectified on F:TKanCplxF : T \to KanCplx. And fPSh (,1)(C op)f \in PSh_{(\infty,1)}(C^{op}) belongs to Alg (,1)(C)Alg_{(\infty,1)}(C) if FF preserves finite products weakly in that for {c iC}\{c_i \in C\} a finite collection of objects, the canonical natural morphism

F(c 1××c n)F(c 1)××F(c n) F(c_1 \times \cdots\times \c_n) \to F(c_1) \times \cdots \times F(c_n)

is a homotopy equivalence of Kan complexes.

If TT is an ordinary category with products, hence an ordinary Lawvere theory, then such a functor is called a homotopy T-algebra. There is a model category structure on these (see there).

We now look at model category structure on strictly product preserving functors CsSetC \to sSet, which gives an equivalent model for Alg (,1)(C)Alg_{(\infty,1)}(C). See model structure on simplicial T-algebras.


Let CC be a category with finite products, and let sTAlgFunc(C,sSet)sTAlg \subset Func(C,sSet) be the full subcategory of the functor category from CC to sSet on those functors that preserve these products.

Then sAlg(C)sAlg(C) carries the structure of a model category sAlg(C) projsAlg(C)_{proj} where the weak equivalences and the fibrations are objectwise those in the standard model structure on simplicial sets.

This is due to (Quillen).

The inclusion i:sAlg(C)sPSh(C op) proji : sAlg(C) \hookrightarrow sPSh(C^{op})_{proj} into the projective model structure on simplicial presheaves evidently preserves fibrations and acylclic fibrations and gives a Quillen adjunction

sAlg(C) projisPSh(C op). sAlg(C)_{proj} \stackrel{\leftarrow}{\underset{i}{\hookrightarrow}} sPSh(C^{op}) \,.

The total right derived functor

i:Ho(sAlg(C) proj)Ho(sPSh(C op) proj) \mathbb{R}i : Ho(sAlg(C)_{proj}) \to Ho(sPSh(C^{op})_{proj})

is a full and faithful functor and an object FsPSh(C op)F \in sPSh(C^{op}) belongs to the essential image of i\mathbb{R}i precisely if it preserves products up to weak homotopy equivalence.

This is due to (Bergner).

It follows that the natural (,1)(\infty,1)-functor

(sAlg(C) proj) PSh (,1)(C op) (sAlg(C)_{proj})^\circ \stackrel{}{\to} PSh_{(\infty,1)}(C^{op})

is an equivalence.

A comprehensive statement of these facts is in HTT, section 5.5.9.


Simplicial 1-algebras

For TT (the syntactic category of) an ordinary algebraic theory (a Lawvere theory) let TAlgT Alg be the category of its ordinary algebras, the ordinary product-preserving functors TSetT \to Set.

We may regard TT as an (,1)(\infty,1)-category and consider its (,1)(\infty,1)-algebras. By the above discussion, these are modeled by product-presering functors TsSetT \to sSet. But this are equivalently simplicial objects in TT-algebras

[T,sSet] ×TAlg Δ op. [T, sSet]_\times \simeq T Alg^{\Delta^{op}} \,.

There is a standard model structure on simplicial T-algebras and we find that simplicial TT-1-algebras model TT-(,1)(\infty,1)-algebras.

Homotopy TT-algebras

For TT an ordinary Lawvere theory, there is also a model category structure on ordinary functors TsSetT \to sSet that preserve the products only up to weak equivalence. Such functors are called homotopy T-algebras.

This model structure is equivalent to the model structure on simplicial T-algebras (see homotopy T-algebra for details) but has the advantage that it is a left proper model category.

Simplicial theories

There is a notion of simplicial algebraic theory that captures some class of (,1)(\infty,1)-algebraic theories. For the moment see section 4 of (Rezk)


A pregeometry (for structured (∞,1)-toposes) is a (multi-sorted) (,1)(\infty,1)-algebraic theory. A structure (,1)(\infty,1)-sheaf on an (∞,1)-topos 𝒳\mathcal{X} in the sense of structured (∞,1)-toposes is an \infty-algebra over this theory

𝒪:𝒯𝒳 \mathcal{O} : \mathcal{T} \to \mathcal{X}

in the (,1)(\infty,1)-topos 𝒳\mathcal{X} – a special one satisfying extra conditions that make it indeed behave like a sheaf of function algebras .

Symmetric monoidal (,1)(\infty,1)-Categories and E E_\infty-algebras

There is a (2,1)(2,1)-algebraic theory whose algebras in (∞,1)Cat are symmetric monoidal (∞,1)-categories. Hence monoids in these algebras are E-∞ algebras (see monoid in a monoidal (∞,1)-category).

This is in (Cranch). For more details see (2,1)-algebraic theory of E-infinity algebras.


The model structure presentation for the (,1)(\infty,1)-category of (,1)(\infty,1)-algebras goes back all the way to

  • Dan Quillen, Homotopical Algebra Lectures Notes in Mathematics 43, SpringerVerlag, Berlin, (1967)

A characterization of (,1)(\infty,1)-categories of (,1)(\infty,1)-algebras in terms of sifted colimits is given in

  • J. Rosicky On homotopy varieties (pdf)

using the incarnation of (,1)(\infty,1)-categories as simplicially enriched categories.

An (,1)(\infty,1)-categorical perspective on these homotopy-algebraic theories is given in

  • Andre Joyal, The theory of quasi-categories and its applications, lectures at CRM Barcelona February 2008, draft hc2.pdf_

from page 44 on.

A detailed account in the context of a general theory of (∞,1)-category of (∞,1)-presheaves is the context of section 5.5.8 of

The model category presentations of (,1)(\infty,1)-algebras is studied in

where it is shown that every such model is Quillen equivalent to a left proper model category. The article uses a monadic definition of (,1)(\infty,1)-algebras.

A discussion of homotopy T-algebras and their strictification is in

  • Bernard Badzioch, Algebraic theories in homotopy theory Annals of Mathematics, 155 (2002), 895-913 (JSTOR)

and for multi-sorted theories in

  • Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007.

A discussion of E-∞ algebra-structures in terms of (,1)(\infty,1)-algebraic theories is in

See also

Expressed as a higher form of Lawvere theory see

Last revised on June 18, 2023 at 15:52:53. See the history of this page for a list of all contributions to it.