model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
equivalences in/of $(\infty,1)$-categories
For $T$ a Lawvere theory and $T Alg$ the category of algebra over a Lawvere theory, there is a model category structure on the category $T Alg^{\Delta^{op}}$ of simplicial $T$-algebras which models the $\infty$-algebras for $T$ regarded as an (∞,1)-algebraic theory.
First we consider the case of simplicial objects in algebras over an ordinary Lawvere theory:
Then we generalize to the case that the Lawvere theory itself is simplicial:
Recall that the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C^{op})$ itself is modeled by the model structure on simplicial presheaves
where we regard $C$ 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 $f : C \to \infty \mathrm{Grp}$ is equivalent to a rectified one $F : C \to KanCplx$. 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
is a homotopy equivalence of Kan complexes.
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 homotopy T-algebras.
Let $T$ be the syntactic category of a Lawvere theory, and let $T Alg^{\Delta^{op}} \subset Func(T,sSet)$ be the full subcategory of the functor category from $T$ to sSet on those functors that preserve these finite products.
Then $T Alg^{\Delta^{op}}$ carries the structure of a model category $(T Alg^{\Delta^{op}})_{proj}$ where the weak equivalences and the fibrations are those maps underlying which are weak equivalences or fibrations, respectively, in the classical model structure on simplicial sets.
This is due to (Quillen 67, II.4 theorem 4).
The inclusion $i : sAlg(C) \hookrightarrow sPSh(C^{op})_{proj}$ into the projective model structure on simplicial presheaves evidently preserves fibrations and acyclic fibrations and gives a Quillen adjunction
The total right derived functor
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 (Badzioch 02, def. 5.2, prop. 5.4, cor. 5.7).
It follows that the natural $(\infty,1)$-functor
is an equivalence.
Let $\Gamma = (Skel(FinSet^{\ast/}))$ be Segal's category, the opposite category of a skeleton of finite pointed sets.
A simplicial Lawvere theory is a a pointed simplicial category $T$ equipped with a functor $i \;\colon\;\Gamma \to T$ such that
$i$ is the identity on objects
$i$ preserves finite products
Given a simplicial theory $T$, then a simplicial $T$-algebra is a product preserving simplicial functor $X$ to the simplicial category of pointed simplicial sets. The simplicial set
(the value on the pointed 2-element set) is called the underlying simplicial set of the $T$-algebra.
A homomorphism of $T$-algebras is a simplicial natural transformation between such functors. Write
for the resulting simplicial category.
A homomorphism is called a weak equivalence or a fibration if on underlying simplicial sets it is a weak equivalence or fibration, respectively, in the classical model structure on simplicial sets. Write
for the category equipped with these classes of morphisms.
(Schwede 01, def. 2.1 and def. 2.2 and beginning of section 3)
In the case that the simplicial Lawvere theory $T$ happens to be simplicially discrete, i.e. an ordinary category, then the category $(T Alg)_{proj}$ of simplicial $T$-algebras from def. with that from prop. .
For $T$ a simplicial Lawvere theory (def. ) the category $(T Alg)_{poj}$ from def. is a simplicial model category.
This is due to (Reedy 74, theorem I), reviewed in (Schwede 01).
The analogous statement with the classical model structure on simplicial sets replaced by the classical model structure on topological spaces is due to (Schwänzl-Vogt 91)
A comprehensive statement of these facts is in HTT, section 5.5.9.
Let $\mathcal{C}$ be a category with enough projectives. Say that a morphism $f$ in the category $\mathcal{C}^{\Delta^{\mathrm{op}}}$ of simplicial objects in $\mathcal{C}$ is a weak equivalence or fibration if for every projective object $P \in \mathcal{C}$ then $Hom(P,f)$ is a weak equivalence or fibration, respectively, in the classical model structure on simplicial sets (i.e. a Kan fibration or weak homotopy equivalence, respectively). If now at least one of the following conditions is satisfied,
every object of $\mathcal{C}^{\Delta^{op}}$ is fibrant;
$\mathcal{C}$ is closed under colimits and has a small set of projective generators;
then this makes $\mathcal{C}^{\Delta^{op}}$ a simplicial model category.
There is a Quillen equivalence between the model structure on simplicial $T$-algebras and the model structure for homotopy T-algebras. (See there).
This is (Badzioch 02, theorem 1.3).
Let $T$ be an abelian Lawvere theory, a theory that contains the theory of abelian group, $Ab \to T$. Then every simplicial $T$-algebra has an underlying simplicial abelian group and is necessarily a Kan complex.
The homotopy groups $\pi_*$ of a simplicial abelian $T$-agebra form an $\mathbb{N}$-graded $T$-algebra $\pi_*(A)$
The inclusion of the full subcategory $i : T Alg \hookrightarrow T Alg^{\Delta^{op}}$ of ordinary $T$-algebra as the simplicially constant ones constitutes a Quillen adjunction
from the trivial model structure on $T Alg$.
The derived functor $\mathbb{R} i : T Alg \to Ho(T Alg^{\Delta^{op}})$ is a full and faithful functor.
This allows us to think of ordinary $T$-algebras a sitting inside $\infty$-$T$-algebras.
all this is certainly true for ordinary $k$-algebras. Need to spell out general proof. I believe this is the Badzioch paper cited above - JB
A simplicial rings is a simplicial $T$-algebras for $T$ the Lawvere theory of rings.
Let $k$ be an ordinary commutative ring and $T$ the theory of commutative associative algebras over $k$. We write $T Alg$ as $sCAlg_k$ or $CAlg_k^{op}$.
Such simplicial $k$-algebras are discussed for instance in (Toën-Vezzosi, section 2.2.1). According to (Schwede 97, Lemma 3.1.3), this model structure is proper.
The following is a variant of the model structure on simplicial commutative algebras that is implied by the above general theorem.
(second model structure)
Let $T$ be the Lawvere theory for (associative and) commutative algebras over a ring $k$. Then $(cAlg_k)^{\Delta^{op}}$ becomes a simplicial model category with
weak equivalences the morphisms whose underlying morphism of simplicial sets are weak equivalences in the classical model structure on simplicial sets;
fibrations the morphisms $X \to Y$ such that $X \to \pi_0 X \times_{\pi_0 Y} Y$ is a degreewise surjection.
This appears as (Goerss-Schemmerhorn, theorem 4.17).
See at model structure on simplicial Lie algebras.
Complete rational Hopf algebras are not the models of a Lawvere theory, even though in some sense they are close (Quillen 67, bottom of p. 265 (61 of 92)).
But they do satisfy the assumptions of proposition (Quillen 69, appendix B, prop. 2.24). Hence simplicial rational complete Hopf algebras form a simplicial model category.
The classical reference for the transferred model structure on simplicial $T$-algebras is
with some further remarks in
The generalization to the case that the theory $T$ itself is allowed to be simplicial is due to
and the topological version is due to
More is in
In
it is discussed that every model category of simplicial $T$-algebras is Quillen equivalent to a left proper model category.
The fact that the model structure on simplicial $T$-algebras serves to model $\infty$-algebras is in
and the multi-sorted version is in
The simplicial model structure on ordinary simplicial algebras (i.e. simplicial associative algebras) is discussed in
Paul Goerss, Kirsten Schemmerhorn, Model categories and simplicial methods (pdf)
Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997), pp. 77-104, pdf.
Discussion of simplicial commutative associative algbras over a ring in the context of derived geometry is in
Discussion of divided power operations on simplicial algebras is in