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Register a new userInternal Neighbourhood Structures III: Finite Sum of Subobjects
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Partha Pratim Ghosh
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The notion of an internal preneighbourhood space on a finitely complete category with finite coproducts and a proper $(mathsf{E}, mathsf{M})$ system such that for each object $X$ the set of $mathsf{M}$-subobjects of $X$ is a complete lattice was initiated in cite{2020}. The notion of a closure operator, closed morphism and its near allies investigated in cite{2021-clos}. The present paper provides structural conditions on the triplet $(mathbb{A}, mathsf{E}, mathsf{M})$ (with $mathbb{A}$ lextensive) equivalent to the set of $mathsf{M}$-subobjects of an object closed under finite sums. Equivalent conditions for the set of closed embeddings (closed morphisms) closed under finite sums is also provided. In case when lattices of admissible subobjects (respectively, closed embeddings) are closed under finite sums, the join semilattice of admissible subobjects (respectively, closed embeddings) of a finite sum is shown to be a biproduct of the component join semilattices. Finally, it is shown whenever the set of closed morphisms is closed under finite sums, the set of proper (respectively, separated) morphisms are also closed under finite sums. This leads to equivalent conditions for the full subcategory of compact (respectively, Hausdorff) preneighbourhood spaces to be closed under finite sums.
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Internal preneighbourhood spaces were first conceived inside any finitely complete category with finite coproducts and proper factorisation structure in my earlier paper. In this paper a closure operation is introduced on internal preneighbourhood spaces and investigated along with closed morphisms and its close allies. Analogues of several well known classes of topological spaces for preneighbourhood spaces are investigated. The approach via preneighbourhood systems is shown to be more general than the closure operators and conveniently allows to identify properties of classes of morphisms which are independent of continuity of morphisms with respect to closure operators.
The main aim of this paper is to provide a description of neighbourhood operators in finitely complete categories with finite coproducts and a proper factorisation system such that the semilattice of admissible subobjects make a distributive complete lattice. The equivalence between neighbourhoods, Kuratowski interior operators and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhoods is shown to be topological. Regular epimorphisms of categories of neighbourhoods are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the category of internal neighbourhoods of topological spaces is the category of bitopological spaces, while in the category of locales every locale comes equipped with a natural internal topology.
When $mathbb C$ is a semi-abelian category, it is well known that the category $mathsf{Grpd}(mathbb C)$ of internal groupoids in $mathbb C$ is again semi-abelian. The problem of determining whether the same kind of phenomenon occurs when the property of being semi-abelian is replaced by the one of being action representable (in the sense of Borceux, Janelidze and Kelly) turns out to be rather subtle. In the present article we give a sufficient condition for this to be true: in fact we prove that the category $mathsf{Grpd}(mathbb C)$ is a semi-abelian action representable algebraically coherent category with normalizers if and only if $mathbb C$ is a semi-abelian action representable algebraically coherent category with normalizers. This result applies in particular to the categories of internal groupoids in the categories of groups, Lie algebras and cocommutative Hopf algebras, for instance.
We develop some basic concepts in the theory of higher categories internal to an arbitrary $infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yonedas lemma for internal categories.
Let $mathscr{C}$ be an additive category with an involution $ast$. Suppose that $varphi : X rightarrow X$ is a morphism of $mathscr{C}$ with core inverse $varphi^{co} : X rightarrow X$ and $eta : X rightarrow X$ is a morphism of $mathscr{C}$ such that $1_X+varphi^{co}eta$ is invertible. Let $alpha=(1_X+varphi^{co}eta)^{-1},$ $beta=(1_X+etavarphi^{co})^{-1},$ $varepsilon=(1_X-varphivarphi^{co})etaalpha(1_X-varphi^{co}varphi),$ $gamma=alpha(1_X-varphi^{co}varphi)beta^{-1}varphivarphi^{co}beta,$ $sigma=alphavarphi^{co}varphialpha^{-1}(1_X-varphivarphi^{co})beta,$ $delta=beta^{ast}(varphi^{co})^{ast}eta^{ast}(1_X-varphivarphi^{co})beta.$ Then $f=varphi+eta-varepsilon$ has a core inverse if and only if $1_X-gamma$, $1_X-sigma$ and $1_X-delta$ are invertible. Moreover, the expression of the core inverse of $f$ is presented. Let $R$ be a unital $ast$-ring and $J(R)$ its Jacobson radical, if $ain R^{co}$ with core inverse $a^{co}$ and $jin J(R)$, then $a+jin R^{co}$ if and only if $(1-aa^{co})j(1+a^{co}j)^{-1}(1-a^{co}a)=0$. We also give the similar results for the dual core inverse.
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