natural homotopy



Topos theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory


Homotopy Theory

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



Paths and cylinders

Homotopy groups

Basic facts




Viewing toposes as generalized spaces, natural homotopy is an equivalence relation between geometric morphisms generalizing the homotopy relation between continuous maps.

This relation is “natural” in the metaphorical sense of being the most “basic” such relation and in the metonymical sense of being defined via natural transformations.

The correspondance between geometric theories and (classifying) Grothendieck toposes then induces an equivalence relation between theories finer than Morita equivalence that affords to define “homotopy theoretical” concepts for logic, e.g. contractability of a theory.



Given two Grothendieck toposes ,\mathcal{E},\mathcal{F}, let [,][\mathcal{E},\mathcal{F}] denote the class of connected components of their hom-category Hom(,)Hom(\mathcal{E},\mathcal{F}) in the 2-category GrTopGrTop. Two geometric morphisms f,g:f,g:\mathcal{E}\to\mathcal{F} are called naturally homotopic , denoted fgf\sim g, if [f]=[g][f]=[g], or in other words, if there exists a zigzag of natural transformations f *g *f^*\leftarrow\cdot\rightarrow\cdot \dots\cdot\leftarrow\cdot\rightarrow g^*.

Remark. Under the identification Hom(,Set(𝕋))𝕋Mod()Hom(\mathcal{E},Set(\mathbb{T}))\cong \mathbb{T}-Mod(\mathcal{E}) natural homotopy corresponds to an equivalence relation \sim on 𝕋\mathbb{T}-models in \mathcal{E} given by zigzags of model homomorphisms.

Properties and some homotopic concepts

Using the Godement calculus of natural transformations an induction over the number of zigzags \cdot\leftarrow\cdot\rightarrow\cdot shows that fff\sim f' implies hfkhfkh\circ f\circ k\sim h\circ f'\circ k. From this it follows that for composable geometric morphisms fff\sim f', ggg\sim g' implies fgfgf\circ g\sim f'\circ g'.

In particular, one can define a (partial) composition [f][g][f]\circ[g] on natural homotopy equivalence classes via the composition of the representatives [fg][f\circ g]. In this way, one obtains a homotopy category Ho(GrTop)\mathbf{Ho}(GrTop) with objects the Grothendieck toposes and morphisms the natural homotopy equivalence classes of geometric morphisms.


A geometric morphism f:f:\mathcal{E}\to\mathcal{F} is called a (natural) homotopy equivalence if there exists a geometric morphism g:g:\mathcal{F}\to\mathcal{E} such that fgid f\circ g\sim id_{\mathcal{F}} and gfid g\circ f\sim id_{\mathcal{E}}. In that case, \mathcal{F} and \mathcal{E} are said to be of the same (natural) homotopy type.

Homotopy equivalences are closed under composition hence “being of the same homotopy type” is an equivalence relation on toposes. Toposes having the same homotopy type as SetSet, the “point” in GrTopGrTop, deserve to be singled out:


A Grothendieck topos Set(𝕋)Set(\mathbb{T})\;, respectively the geometric theory 𝕋\mathbb{T} it classifies, is called (naturally) contractible , if there exists a “constant” geometric morphism c:Set(𝕋)Set(𝕋)c:Set(\mathbb{T})\to Set(\mathbb{T}) i.e. one factoring through SetSet as Set(𝕋)!Setc¯Set(𝕋)Set(\mathbb{T})\overset{!}{\to}Set\overset{\bar{c}}{\to}Set(\mathbb{T})\;, such that cid Set(𝕋)c\sim id_{Set(\mathbb{T})}\;.

In other words, a topos is contractible if its identity map is homotopic to a constant map and a geometric theory 𝕋\mathbb{T} is contractible iff its classifying topos Set(𝕋)Set(\mathbb{T}) is.

Notice incidentally, that this requires by definition the existence of a point of Set(𝕋)Set(\mathbb{T}), or in other words, the existence of a “classical” i.e. SetSet-based 𝕋\mathbb{T}-model. In particular, since there exist (Boolean) toposes without points there exist (Boolean) geometric theories that are not contractible.

The correspondance between geometric theories and Grothendieck toposes together with the correspondance between their models and geometric morphisms similarly affords the transposition of other homotopic concepts to the realm of geometric logic.

Let us take a closer look at constant endomorphisms cc of Set(𝕋)Set(\mathbb{T})\;:

Set(𝕋)!Setc¯Set(𝕋). Set(\mathbb{T})\overset{!}{\to}Set\overset{\bar{c}}{\to}Set(\mathbb{T})\;.

Here c¯\bar{c} classifies a 𝕋\mathbb{T}-model in SetSet whereas !! classifies a model (which is unique up to isomorphism since SetSet is terminal) of some theory 𝕊\mathbb{S} classified by SetSet whence cc classifies a 𝕋\mathbb{T}-model that carries a 𝕊\mathbb{S}-model structure as well.

Now, the inverse image part of !! is Δ\Delta\;, the constant sheaf functor. Objects in the range of Δ\Delta are traditionally called “constant” whence we can think of the contractability id Set(𝕋)cid_{Set(\mathbb{T})}\sim c of 𝕋\mathbb{T} as expressing the “almost constancy” of the generic model M 𝕋Δ(c¯ *(M 𝕋))\mathbf{M}_\mathbb{T}\sim \Delta(\bar{c}^*(\mathbf{M}_\mathbb{T})) i.e. as a structural similarity between M 𝕋\mathbf{M}_\mathbb{T} and a constant 𝕋\mathbb{T}-model.


Let Set(𝔻 )Set(\mathbb{D}_\infty) be the Schanuel topos that classifies the theory 𝔻 \mathbb{D}_\infty of infinite decidable objects and D \mathbf{D}_\infty the generic infinite decidable object.

Proposition (Joyal-Wraith)

Set(𝔻 )Set(\mathbb{D}_\infty)\;, respectively the theory 𝔻 \mathbb{D}_\infty\;, are naturally contractible.


We argue on the logical side: SetSet classifies the theory 𝕊\mathbb{S} of standard successor algebras whose models are natural number objects (cf. the example section at geometric theory). But by generalities (that can be found e.g. in Borceux vol.3) a natural number object is both infinite and decidable and certainly the familiar set \mathbb{N} of natural numbers in Set is.

Accordingly, we take c¯\bar{c} as the map that classifies =c¯ *(D )\mathbb{N}=\bar{c}^*(\mathbf{D}_\infty) as a model of 𝔻 \mathbb{D}_\infty in Set. Then !=ΔΓ!=\Delta\dashv\Gamma classifies Δ()\Delta(\mathbb{N}) as a model of 𝕊\mathbb{S} in Set(𝔻 )Set(\mathbb{D}_\infty) and the composite c¯!\bar{c}\circ ! classifies Δ()=Δ(c¯ *(D ))\Delta(\mathbb{N})=\Delta(\bar{c}^*(\mathbf{D}_\infty)) as a model of 𝔻 \mathbb{D}_\infty in Set(𝔻)Set(\mathbb{D}).

Whence it suffices to exhibit a zigzag of homomorphisms in 𝔻 Mod(Set(𝔻 ))\mathbb{D}_\infty-Mod(Set(\mathbb{D}_\infty)) between the natural numbers object Δ()\Delta(\mathbb{N}) and the generic model D \mathbf{D}_\infty but the coproduct Δ()+D \Delta(\mathbb{N})+\mathbf{D}_\infty is both infinite and decidable, hence in 𝔻 Mod(Set(𝔻 ))\mathbb{D}_\infty-Mod(Set(\mathbb{D}_\infty))\;, and the respective inclusions Δ()Δ()+D D \Delta(\mathbb{N})\rightarrow \Delta(\mathbb{N})+\mathbf{D}_\infty\leftarrow \mathbf{D}_\infty yield the desired zigzag of homomorphisms.\qed

Topos homotopy via path spaces

The Sierpinski topos Set 2Set^2 is a connected, locally connected and local topos. As a result of Artin gluing along SetidSetSet\overset{id}{\to}Set it has two topos points (corresponding to the two points of the underlying Sierpinski space). It is exponentiable and, given a topos \mathcal{E}, we can accordingly view the exponential Set 2\mathcal{E}^{Set^2} as a path space for \mathcal{E} with respect to the abstract interval object Set 2Set^2.

(Cf. Beke 2000, p.11f)


The concept originates with

  • André Joyal, Gavin Wraith, Eilenberg-Mac Lane Toposes and Cohomology , pp.117-131 in Cont. Math. 92 AMS 1984.

  • Gavin Wraith, Toposes and simplicial sets: the cohomological connection , pp.281-290 in Category theoretic methods in geometry , no.35 Aarhus Univ. Var. Publ. Ser. 1983.

An overview of homotopy theory done in this “toposophical” context is

  • Tibor Beke, Homotopoi , ms. University of Massachusetts Lowell (2000). (dvi)

The following paper explores the topos of sheaves on the unit interval as abstract interval object

  • Ieke Moerdijk, Gavin Wraith, Connected locally connected toposes are path-connected , Trans. AMS 295 no.2 (1986) pp.849-859.

A proof of the homotopy equivalence of certain petit and gros toposes can be found on p.415f in

A good source for the classical cohomology theory of toposes is chapter 8 of

  • Peter Johnstone, Topos Theory , Academic Press New York (1977). (Also available as Dover Reprint, Mineola 2014)

Last revised on October 13, 2020 at 06:34:39. See the history of this page for a list of all contributions to it.