homotopy theory, (∞,1)-category theory, homotopy type theory
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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 $Hom(\mathcal{E},\mathcal{F})$ in the 2-category $GrTop$. Two geometric morphisms $f,g:\mathcal{E}\to\mathcal{F}$ are called naturally homotopic , denoted $f\sim g$, if $[f]=[g]$, or in other words, if there exists a zigzag of natural transformations $f^*\leftarrow\cdot\rightarrow\cdot \dots\cdot\leftarrow\cdot\rightarrow g^*$.
Remark. Under the identification $Hom(\mathcal{E},Set(\mathbb{T}))\cong \mathbb{T}-Mod(\mathcal{E})$ natural homotopy corresponds to an equivalence relation $\sim$ on $\mathbb{T}$-models in $\mathcal{E}$ given by zigzags of model homomorphisms.
Using the Godement calculus of natural transformations an induction over the number of zigzags $\cdot\leftarrow\cdot\rightarrow\cdot$ shows that $f\sim f'$ implies $h\circ f\circ k\sim h\circ f'\circ k$. From this it follows that for composable geometric morphisms $f\sim f'$, $g\sim g'$ implies $f\circ g\sim f'\circ g'$.
In particular, one can define a (partial) composition $[f]\circ[g]$ on natural homotopy equivalence classes via the composition of the representatives $[f\circ g]$. In this way, one obtains a homotopy category $\mathbf{Ho}(GrTop)$ with objects the Grothendieck toposes and morphisms the natural homotopy equivalence classes of geometric morphisms.
A geometric morphism $f:\mathcal{E}\to\mathcal{F}$ is called a (natural) homotopy equivalence if there exists a geometric morphism $g:\mathcal{F}\to\mathcal{E}$ such that $f\circ g\sim id_{\mathcal{F}}$ and $g\circ f\sim id_{\mathcal{E}}$. 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 $Set$, the “point” in $GrTop$, deserve to be singled out:
A Grothendieck topos $Set(\mathbb{T})\;$, respectively the geometric theory $\mathbb{T}$ it classifies, is called (naturally) contractible , if there exists a “constant” geometric morphism $c:Set(\mathbb{T})\to Set(\mathbb{T})$ i.e. one factoring through $Set$ as $Set(\mathbb{T})\overset{!}{\to}Set\overset{\bar{c}}{\to}Set(\mathbb{T})\;$, such that $c\sim id_{Set(\mathbb{T})}\;$.
In other words, a topos is contractible if its identity map is homotopic to a constant map and a geometric theory $\mathbb{T}$ is contractible iff its classifying topos $Set(\mathbb{T})$ is.
Notice incidentally, that this requires by definition the existence of a point of $Set(\mathbb{T})$, or in other words, the existence of a “classical” i.e. $Set$-based $\mathbb{T}$-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 $c$ of $Set(\mathbb{T})\;$:
Here $\bar{c}$ classifies a $\mathbb{T}$-model in $Set$ whereas $!$ classifies a model (which is unique up to isomorphism since $Set$ is terminal) of some theory $\mathbb{S}$ classified by $Set$ whence $c$ classifies a $\mathbb{T}$-model that carries a $\mathbb{S}$-model structure as well.
Now, the inverse image part of $!$ is $\Delta\;$, the constant sheaf functor. Objects in the range of $\Delta$ are traditionally called “constant” whence we can think of the contractability $id_{Set(\mathbb{T})}\sim c$ of $\mathbb{T}$ as expressing the “almost constancy” of the generic model $\mathbf{M}_\mathbb{T}\sim \Delta(\bar{c}^*(\mathbf{M}_\mathbb{T}))$ i.e. as a structural similarity between $\mathbf{M}_\mathbb{T}$ and a constant $\mathbb{T}$-model.
Let $Set(\mathbb{D}_\infty)$ be the Schanuel topos that classifies the theory $\mathbb{D}_\infty$ of infinite decidable objects and $\mathbf{D}_\infty$ the generic infinite decidable object.
$Set(\mathbb{D}_\infty)\;$, respectively the theory $\mathbb{D}_\infty\;$, are naturally contractible.
Proof.
We argue on the logical side: $Set$ classifies the theory $\mathbb{S}$ 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 $\bar{c}$ as the map that classifies $\mathbb{N}=\bar{c}^*(\mathbf{D}_\infty)$ as a model of $\mathbb{D}_\infty$ in Set. Then $!=\Delta\dashv\Gamma$ classifies $\Delta(\mathbb{N})$ as a model of $\mathbb{S}$ in $Set(\mathbb{D}_\infty)$ and the composite $\bar{c}\circ !$ classifies $\Delta(\mathbb{N})=\Delta(\bar{c}^*(\mathbf{D}_\infty))$ as a model of $\mathbb{D}_\infty$ in $Set(\mathbb{D})$.
Whence it suffices to exhibit a zigzag of homomorphisms in $\mathbb{D}_\infty-Mod(Set(\mathbb{D}_\infty))$ between the natural numbers object $\Delta(\mathbb{N})$ and the generic model $\mathbf{D}_\infty$ but the coproduct $\Delta(\mathbb{N})+\mathbf{D}_\infty$ is both infinite and decidable, hence in $\mathbb{D}_\infty-Mod(Set(\mathbb{D}_\infty))\;$, and the respective inclusions $\Delta(\mathbb{N})\rightarrow \Delta(\mathbb{N})+\mathbf{D}_\infty\leftarrow \mathbf{D}_\infty$ yield the desired zigzag of homomorphisms.$\qed$
The Sierpinski topos $Set^2$ is a connected, locally connected and local topos. As a result of Artin gluing along $Set\overset{id}{\to}Set$ 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 $\mathcal{E}^{Set^2}$ as a path space for $\mathcal{E}$ with respect to the abstract interval object $Set^2$.
(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
The following papers explore 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.
Ieke Moerdijk, Path-lifting for Grothendieck toposes, Proc. AMS 102 no.2 (1988) 2424-248 $[$doi:10.2307/2045869$]$
Ieke Moerdijk, An addendum to “Path-lifting for Grothendieck toposes”, Proc. AMS 125 no.9 (1997) 2815-2818 $[$doi:2162060$]$
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
Last revised on June 14, 2022 at 10:08:30. See the history of this page for a list of all contributions to it.