nLab (∞,1)-algebraic theory

Idea

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.

Definition

Definition

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

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

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

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

In a full $(\infty,1)$-category theoretic context this appears as HTT, def. 5.5.8.8. A definition in terms of simplicially enriched categories and the model structure on sSet-categories to present $(\infty,1)$-categories is in Ros. The introduction of that article lists further and older occurences of this definition.

Properties

Proposition

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

• $Alg_{(\infty,1)}(C)$ is an accessible localization of the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C^{op})$ (on the opposite).

  So in particular it is a locally presentable (∞,1)-category.

• $Alg_{(\inft)}$ is a compactly generated (∞,1)-category.

• The $(\infty,1)$-Yoneda embedding $j : C^{op} \to PSh_{(\infty,1)}(C^{op})$ factors through $Alg_{(\infty,1)}(C)$.

• The full subcategory $Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)$ is stable under sifted colimits.

This is HTT, prop. 5.5.8.10.

Models

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

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

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

where we regard $T$ 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 $(\infty,1)$-functor $f : T \to \infty \mathrm{Grp}$ is equivalent to a rectified on $F : T \to KanCplx$. And $f \in PSh_{(\infty,1)}(C^{op})$ belongs to $Alg_{(\infty,1)}(C)$ if $F$ preserves finite products weakly in that for $\{c_i \in C\}$ a finite collection of objects, the canonical natural morphism

$$F(c_1 \times \cdots\times \c_n) \to F(c_1) \times \cdots \times F(c_n)$$

is a homotopy equivalence of Kan complexes.

If $T$ 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 $C \to sSet$, which gives an equivalent model for $Alg_{(\infty,1)}(C)$. See model structure on simplicial T-algebras.

Proposition

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

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

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

Proposition

The total right derived functor

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

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

This is due to (Bergner).

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

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

is an equivalence.

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

Examples

Simplicial 1-algebras

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

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

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

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

Homotopy $T$-algebras

For $T$ an ordinary Lawvere theory, there is also a model category structure on ordinary functors $T \to sSet$ 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 $(\infty,1)$-algebraic theories. For the moment see section 4 of (Rezk)

Structure-$(\infty,1)$-sheaves

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

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

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

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

There is a $(2,1)$-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.

Related concepts

• algebraic theory / Lawvere theory / essentially algebraic theory

  • 2-Lawvere theory

  • algebraic $(\infty,1)$-theory / essentially algebraic (∞,1)-theory

    • simplicial Lawvere theory

• monad / (∞,1)-monad

• operad / (∞,1)-operad

References

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

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

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

• J. Rosicky On homotopy varieties (pdf)

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

An $(\infty,1)$-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

• Jacob Lurie, Higher Topos Theory .

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

• Charles Rezk, Every homotopy theory of simplicial algebras admits a proper model (math/0003065) ,

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

• James Cranch, Algebraic Theories and $(\infty,1)$-Categories (arXiv)

See also

• MO discussion

Expressed as a higher form of Lawvere theory see

• John D. Berman, Higher Lawvere theories (arXiv:1903.02991)