An essential, hyperconnected, local geometric morphism that is not locally connected
aa r X i v : . [ m a t h . C T ] S e p An essential, hyperconnected, local geometric morphismthat is not locally connected
Jens Hemelaer ∗ Morgan Rogers † Abstract
We give an example of an essential, hyperconnected, local geometric morphismthat is not locally connected, arising from our work-in-progress on geometric mor-phisms
PSh ( M ) → PSh ( N ), where M and N are monoids. Thomas Streicher asked on the category theory mailing list whether every essential,hyperconnected, local geometric morphism is automatically locally connected. We showthat this is not the case, by providing a counterexample. Our counterexample arises fromour earlier work [1] and work-in-progress regarding properties of geometric morphisms
PSh ( M ) → PSh ( N ) for monoids M and N .We thank Thomas Streicher for the interesting question and the subsequent discus-sion regarding this counterexample.The second named author was supported in this work by INdAM and the MarieSklodowska-Curie Actions as a part of the INdAM Doctoral Programme in Mathematicsand/or Applications Cofunded by Marie Sklodowska-Curie Actions . The counterexample
Let M be the monoid with presentation h e, x : e = e, xe = x i . Note that each elementof M can be written as either x n or ex n for some n ∈ { , , , . . . } . Further, let N be thefree monoid on one variable a , so N = { , a, a , . . . } . Consider the monoid morphism φ : M → N which on generators is given by φ ( e ) = 1 and φ ( x ) = a . If we interpret M and N as categories, then φ is a functor. There is an induced essential geometricmorphism PSh ( M ) PSh ( N ) f ∗ Department of Mathematics, University of Antwerp, Middelheimlaan 1, B-2020 Antwerp (Belgium)email: [email protected] † Universit`a degli Studi dell’Insubria, Via Valleggio n. 11, 22100 Como COMarie Sklodowska-Curie fellow of the Istituto Nazionale di Alta Matematicaemail: [email protected]
PSh ( M ) PSh ( N ) f ∗ f ! f ∗ with the following description, for X in PSh ( M ) and Y in PSh ( N ): • f ! ( X ) ≃ X ⊗ M N where N has left M -action defined by m · n = φ ( m ) n for m ∈ M and n ∈ N , and right N -action defined by multiplication; • f ∗ ( Y ) ≃ Y with right M -action defined as y · m = y · φ ( m ) for y ∈ Y and m ∈ M ; • f ∗ ( Y ) ≃ H om M ( N, Y ), where N has right M -action given by n · m = nφ ( m ) for n ∈ N and m ∈ M , and H om M ( N, Y ) is the set of morphisms of right M -sets g : N → Y ; the right N -action on H om M ( N, Y ) is defined as ( g · n )( n ′ ) = g ( nn ′ )for g ∈ H om M ( N, Y ) and n, n ′ ∈ N .For definitions and background regarding tensor products and Hom-sets, in the con-text of sets with a monoid action, we refer to [1, Subsection 1.2]. Proposition 1.
The geometric morphism f is hyperconnected.Proof. This follows from φ being surjective, see [2, Example A.4.6.9]. Proposition 2.
The geometric morphism f is local.Proof. First, we claim that f ∗ has a right adjoint. By the Special Adjoint FunctorTheorem, it is enough to show that f ∗ preserves colimits. This is equivalent to thefunctor H om M ( N, − ) : PSh ( M ) → Set preserving colimits, since colimits in
PSh ( N )are computed on underlying sets. By [5, Lemma 4.1] this is equivalent to N being anindecomposable projective right M -set. To see that N is an indecomposable projectiveright M -set, consider the map φ : M → N . It is a morphism of right M -sets, and ithas a section σ : N → M defined by σ ( a n ) = ex n . The function σ is a morphism ofright M -sets, so N is a retract of M . Since M is indecomposable projective, and N is aretract of M , we have that N is indecomposable projective as well.In order for f to be local, we additionally need that f is connected. But this followsfrom f being hyperconnected, see Proposition 1.Suppose that f is locally connected. Then it follows from the above and [3, Propo-sition 3.5] that f ! preserves binary products. We will show that this is not the case, andas a result f is not locally connected. Proposition 3.
The functor f ! does not preserve binary products.Proof. Consider the right M -sets M and N , with the right M -action on M given bymultiplication, and the right M -action on N given by n · m = nφ ( m ). We claim thatthe natural comparison map f ! ( M × N ) −→ f ! ( M ) × f ! ( N )2s not injective. For a general right M -set X , we have f ! ( X ) ≃ X ⊗ M N , where the left M -action on N is given by m · n = φ ( m ) n . The right N -action on X ⊗ M N is givenby ( x ⊗ n ) · n ′ = x ⊗ nn ′ . In particular, f ! ( M ) ≃ M ⊗ M N ≃ N . Further, the map N ⊗ M N → N , n ⊗ n ′ nn ′ is an isomorphism, so f ! ( N ) ≃ N as well.Using the above, we can rewrite the comparison map as α : ( M × N ) ⊗ M N −→ N × N ( m, n ) ⊗ n ′ ( φ ( m ) n ′ , nn ′ )Note that α (( x, ⊗
1) = ( a,
1) = α (( ex, ⊗ x, ⊗ = ( ex, ⊗ α is not injective. First we show that ( x,
1) and ( ex,
1) are in differentcomponents of M × N . Write: A ′ = { ( x, , ( x , a ) , ( x , a ) , ( x , a ) , . . . } A ′′ = { ( ex, , ( ex , a ) , ( ex , a ) , ( ex , a ) , . . . } B = { ( m, n ) ∈ M × N : φ ( m ) = an } It can be checked directly that A ′ and A ′′ are sub- M -sets of M × N . From cancellativityof N it follows that B ⊆ M × N is a sub- M -set as well. So we can write M × N as adirect sum (coproduct) of right M -sets M × N = A ′ ⊔ A ′′ ⊔ B. We have ( x, ∈ A ′ and ( ex, ∈ A ′′ . Because f ! preserves colimits, we find that( M × N ) ⊗ M N ≃ f ! ( M × N ) ≃ f ! ( A ′ ) ⊔ f ! ( A ′′ ) ⊔ f ! ( B ) . Since ( x, ⊗ ∈ f ! ( A ′ ) and ( ex, ⊗ ∈ f ! ( A ′′ ), we have that ( x, ⊗ = ( ex, ⊗ α is not injective.From the discussion before Proposition 3 it now follows that: Corollary 4.
The geometric morphism f is not locally connected. Bibliography [1] J. Hemelaer and M. Rogers,
Monoid Properties as Invariants of Toposes of MonoidActions , preprint (2020), arXiv:2004.10513 .[2] P.T. Johnstone,
Sketches of an Elephant: A Topos Theory Compendium , The Claren-don Press, Oxford University Press, Oxford, 2002.[3] ,
Remarks on punctual local connectedness , Theory Appl. Categ. (2011),No. 3, 51–63. MR 2805745[4] F. William Lawvere, Axiomatic cohesion , Theory Appl. Categ. (2007), No. 3,41–49. MR 2369017[5] M. Rogers, Toposes of Discrete Monoid Actions , preprint (2019), arXiv:1905.10277arXiv:1905.10277