Abstract
Thomas Streicher asked on the category theory mailing list whether every essential, hyperconnected, local geometric morphism is automatically locally connected. We show that this is not the case, by providing a counterexample.
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We show that an essential, hyperconnected, local geometric morphism is not necessarily locally connected, by constructing a counterexample. This answers a question asked by Thomas Streicher on the category theory mailing list. Our counterexample arises from our earlier work [1] and workinprogress regarding properties of geometric morphisms \({{\,\mathrm{\mathbf {PSh}}\,}}(M) \rightarrow {{\,\mathrm{\mathbf {PSh}}\,}}(N)\) for monoids M and N.
We thank Thomas Streicher for the interesting question and the subsequent discussion regarding this counterexample. Further, we would like to thank the anonymous reviewer for their suggestions, especially for the different proof that f is not locally connected (which replaces our previous, longer proof).
The first named author is a postdoctoral fellow of the Research Foundation – Flanders (File number 1276521N). The second named author was supported in this work by INdAM and the Marie SklodowskaCurie Actions as a part of the INdAM Doctoral Programme in Mathematics and/or Applications Cofunded by Marie SklodowskaCurie Actions.
1 The counterexample
Let M be the monoid with presentation \(\langle {e,x : e^2 = e, xe = x}\rangle \). Note that each element of M can be written as either \(x^n\) or \(ex^n\) for some \(n \in \{0,1,2,\dots \}\). Further, let N be the free monoid on one variable a, so \(N = \{1, a, a^2,\dots \}\). Consider the monoid morphism \(\phi : M \rightarrow N\) which on generators is given by \(\phi (e)=1\) and \(\phi (x) = a\). If we interpret M and N as categories, then \(\phi \) is a functor. There is an induced essential geometric morphism
given by functors
with the following description, for X in \({{\,\mathrm{\mathbf {PSh}}\,}}(M)\) and Y in \({{\,\mathrm{\mathbf {PSh}}\,}}(N)\):

\(f_!(X) \simeq X \otimes _M N\) where N has left Maction defined by \(m \cdot n = \phi (m)n\) for \(m \in M\) and \(n \in N\), and right Naction defined by multiplication;

\(f^*(Y) \simeq Y\) with right Maction defined as \(y \cdot m = y \cdot \phi (m)\) for \(y \in Y\) and \(m \in M\);

\(f_*(X) \simeq \mathcal {H}\! om _M(N,X)\), where N has right Maction given by \(n \cdot m = n\phi (m)\) for \(n \in N\) and \(m \in M\), and \(\mathcal {H}\! om _M(N,X)\) is the set of morphisms of right Msets \(g: N \rightarrow Y\); the right Naction on \(\mathcal {H}\! om _M(N,X)\) is defined as \((g \cdot n)(n') = g(nn')\) for \(g \in \mathcal {H}\! om _M(N,X)\) and \(n,n' \in N\).
For definitions and background regarding tensor products and Homsets, in the context of sets with a monoid action, we refer to [1, Subsection 1.2].
Proposition 1
The geometric morphism f is hyperconnected and local.
Proof
Because \(\phi \) is surjective, it follows that f is hyperconnected, see [2, Example A.4.6.9]. We now show that f is local. Because f is connected (even hyperconnected), it follows from [3, Corollary 3.3] that f is local if and only if \(f_*\) has a further right adjoint \(f^!\). Note that there is an isomorphism of right Msets \(N \cong eM\), so \(f_*(X) \simeq \mathcal {H}\! om _M(eM,X)\). A map of right Msets \(eM \rightarrow X\) is completely determined by the image of e, and this image can be any element of Xe. So we see that \(f_*(X) \simeq Xe\), with the right Naction defined as \(b \cdot a = b\cdot x\) for \(b \in Xe\). In other words, \(f_*(X) \simeq X \otimes _M Me\), where Me has a left Maction by multiplication and a right Naction given by \(m \cdot a = m \cdot x\) for \(m \in Me\). From the tensor–hom adjunction (see e.g. [1, Proposition 1.5]), we now know that \(f_*\) has a right adjoint given by \(f^!(Y) \simeq \mathcal {H}\! om _N(Me,Y)\) for Y in \(\mathbf {PSh}(N)\). It follows that f is local.\(\square \)
Remark 2
As suggested by the reviewer, we can give an explicit description of the functor \(f^!\) as follows. We define \(f^!(Y) \simeq Y \times Y\), with the right Maction defined as
for \((s,t) \in Y \times Y\). To see that this agrees with the description \(f^!(Y)\simeq \mathcal {H}\! om _N(Me,Y)\) as above, note that \(Me \cong N \sqcup N\) as right Nset. The right Maction on
is the one induced by the left Maction on Me (by multiplication).
If f were locally connected, then in particular \(f^*\) would preserve exponential objects. We will show that this is not the case, and as a result f is not locally connected.
Proposition 3
The functor \(f^*\) does not preserve exponential objects.
Proof
Let \(2 = 1 \sqcup 1\). We claim that the comparison map
is not an isomorphism. In our case, the comparison map is the map
where \(g(m,n) = (\phi (m),n)\). We can identify \(\mathcal {H}\! om _N(N\times N,2)\) with the set of complemented subNsets of \(N \times N\), and similarly we can identify \(\mathcal {H}\! om _M(M\times N,2)\) with the set of complemented subMsets of \(M \times N\). The comparison map then sends \(S\subseteq N \times N\) to \(g^{1}(S)\subseteq M\times N\). Now consider the (right) subMset
To verify that this is a complemented subset, note that there is a degree function
such that \(\deg (y \cdot m) = \deg (y)\) for all \(y \in M \times N\), \(m \in M\). Further, if \(y \in M \times N\) is of the form \(y = (x^{n+1},a^n)\), then both \(y \cdot x = (x^{n+2},a^{n+1})\) and \(y \cdot e = y\) are again of this form. So T is complemented, but it is not of the form \(g^{1}(S)\) for some \(S \subseteq N \times N\), because it contains the element (ex, 1) but not the element (x, 1). So the comparison map is not surjective, and as a result \(f^*\) does not preserve exponential objects. \(\square \)
Corollary 4
The geometric morphism f is not locally connected.
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References
Hemelaer, J., Rogers, M.: Monoid properties as invariants of toposes of monoid actions. Appl. Categ. Struct. (2020). https://doi.org/10.1007/s1048502009620y
Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium. The Clarendon Press, Oxford University Press, Oxford (2002)
Johnstone, P.T.: Remarks on punctual local connectedness. Theory Appl. Categ. 25(3), 51–63 (2011)
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Communicated by Thomas Streicher.
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Hemelaer, J., Rogers, M. An Essential, Hyperconnected, Local Geometric Morphism that is not Locally Connected. Appl Categor Struct 29, 573–576 (2021). https://doi.org/10.1007/s10485020096266
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DOI: https://doi.org/10.1007/s10485020096266