# nLab (∞,1)-algebraic theory

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

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

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

${\mathrm{Alg}}_{\left(\infty ,1\right)}\left(C\right)↪{\mathrm{PSh}}_{\left(\infty ,1\right)}\left({C}^{\mathrm{op}}\right)$Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})

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

In a full $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-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

• ${\mathrm{Alg}}_{\left(\infty ,1\right)}\left(C\right)$ is an accessible localization of the (∞,1)-category of (∞,1)-presheaves ${\mathrm{PSh}}_{\left(\infty ,1\right)}\left({C}^{\mathrm{op}}\right)$ (on the opposite).

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

• ${\mathrm{Alg}}_{\left(inft\right)}$ is a compactly generated (∞,1)-category?.

• The $\left(\infty ,1\right)$-Yoneda embedding $j:{C}^{\mathrm{op}}\to {\mathrm{PSh}}_{\left(\infty ,1\right)}\left({C}^{\mathrm{op}}\right)$ factors through ${\mathrm{Alg}}_{\left(\infty ,1\right)}\left(C\right)$.

• The full subcategory ${\mathrm{Alg}}_{\left(\infty ,1\right)}\left(C\right)↪{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)$ is stable under sifted colimits.

This is HTT, prop. 5.5.8.10.

## Models

There are various model category presentations of ${\mathrm{Alg}}_{\left(\infty ,1\right)}\left(C\right)↪{\mathrm{PSh}}_{\left(\infty ,1\right)}\left({C}^{\mathrm{op}}\right)$.

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

${\mathrm{PSh}}_{\left(\infty ,1\right)}\left({C}^{\mathrm{op}}\right)\simeq \left[T,\mathrm{sSet}{\right]}^{\circ }\phantom{\rule{thinmathspace}{0ex}},$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 $\left(-{\right)}^{\circ }$ denoting the full enriched subcategory on fibrant-cofibrant objects.

This says in particular that every weak $\left(\infty ,1\right)$-functor $f:T\to \infty \mathrm{Grp}$ is equivalent to a rectified on $F:T\to \mathrm{KanCplx}$. And $f\in {\mathrm{PSh}}_{\left(\infty ,1\right)}\left({C}^{\mathrm{op}}\right)$ belongs to ${\mathrm{Alg}}_{\left(\infty ,1\right)}\left(C\right)$ if $F$ preserves finite products weakly in that for $\left\{{c}_{i}\in C\right\}$ a finite collection of objects, the canonical natural morphism

$F\left({c}_{1}×\cdots ,{c}_{n}\right)\to F\left({c}_{1}\right)×\cdots ×F\left({c}_{n}\right)$F(c_1 \times \cdots, \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 $C\to \mathrm{sSet}$, which gives an equivalent model for ${\mathrm{Alg}}_{\left(\infty ,1\right)}\left(C\right)$. See model structure on simplicial T-algebras.

###### Proposition

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

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

$\mathrm{sAlg}\left(C{\right)}_{\mathrm{proj}}\stackrel{←}{\underset{i}{↪}}\mathrm{sPSh}\left({C}^{\mathrm{op}}\right)\phantom{\rule{thinmathspace}{0ex}}.$sAlg(C)_{proj} \stackrel{\leftarrow}{\underset{i}{\hookrightarrow}} sPSh(C^{op}) \,.
###### Proposition

The total right derived functor

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

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

This is due to (Bergner).

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

$\left(\mathrm{sAlg}\left(C{\right)}_{\mathrm{proj}}{\right)}^{\circ }\stackrel{}{\to }{\mathrm{PSh}}_{\left(\infty ,1\right)}\left({C}^{\mathrm{op}}\right)$(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 $T\mathrm{Alg}$ be the category of its ordinary algebras, the ordinary product-preserving functors $T\to \mathrm{Set}$.

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

$\left[T,\mathrm{sSet}{\right]}_{×}\simeq T{\mathrm{Alg}}^{{\Delta }^{\mathrm{op}}}\phantom{\rule{thinmathspace}{0ex}}.$[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 $T$-$\left(\infty ,1\right)$-algebras.

### Homotopy $T$-algebras

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

### Structure-$\left(\infty ,1\right)$-sheaves

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

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

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

### Symmetric monoidal $\left(\infty ,1\right)$-Categories and ${E}_{\infty }$-algebras

There is a $\left(2,1\right)$-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.

## References

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

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

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

• J. Rosicky On homotopy varieties (pdf)

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

An $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-algebraic theories is in

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

Revised on December 29, 2010 16:50:47 by Urs Schreiber (89.204.137.120)