---
title: Join Restriction Categories
url: https://www.emergentmind.com/topics/join-restriction-categories
type: topic
---

# Join Restriction Categories

Searching arXiv for recent and foundational papers on join restriction categories and related developments.
Join restriction categories are restriction categories in which compatible partial maps can be glued. More precisely, they refine the axiomatic study of partially defined mappings by requiring that every compatible family of parallel morphisms admit a join, with this gluing operation stable under composition. In the literature, they appear both as an abstract calculus of partiality and as a structural interface with enriched category theory, sheaf theory, distributive and classical restriction structure, internal étale constructions, and tangent-geometric formulations of bundles and Lie theory [1211.6170] [1801.07390].

## 1. Axiomatic core

A restriction category \(\C\) consists of a category equipped with an assignment
\[
\mathrm{Res}\;:\;\Mor(\C)\;\longrightarrow\;\Mor(\C),\quad f\;\mapsto\;\bar f\;:\;X\to X
\]
subject to the axioms, for all composable
\[
\begin{array}{l}
f\colon X\to Y,\;g\colon X\to Z,\;h\colon Y\to Z:
\end{array}
\]
\[
\text{(R1)}\quad f\,\bar f = f,
\qquad
\text{(R2)}\quad \bar f\,\bar g = \bar g\,\bar f,
\]
\[
\text{(R3)}\quad \overline{\,g\,\bar f\,} = \bar g\,\bar f,
\qquad
\text{(R4)}\quad \bar h\,f = f\,\overline{h f}.
\]
Each \(\bar f\) is an idempotent on its domain, called a restriction idempotent. Morphisms with \(\bar f=1\) are total. Each hom-set \(\C(X,Y)\) carries a partial order
\[
f\;\le\;g
\quad:\Longleftrightarrow\quad
g\,\bar f
\;=\;
f\,.
\]

In a restriction category, a family of parallel maps \(\{f_i:X\to Y\mid i\in I\}\) is compatible if
\[
f_i\,\overline{f_j}\;=\;f_j\,\overline{f_i}
\qquad
(\forall\,i,j\in I).
\]
A join restriction category is one in which every compatible family admits a join \(\bigvee_i f_i\) in the order above, and these joins satisfy
\[
\text{(J1)}\quad
\overline{\bigvee_i f_i}
\;=\;
\bigvee_i\,\overline{f_i},
\]
\[
\text{(J2)}\quad
\Bigl(\bigvee_i f_i\Bigr)\,g
\;=\;
\bigvee_i(f_i\,g),
\]
\[
\text{(J3)}\quad
h\,\Bigl(\bigvee_i f_i\Bigr)
\;=\;
\bigvee_i (h\,f_i).
\]
Guo–Kock–Cockett et al. show that \((\mathrm{J3})\) follows from \((\mathrm{J1})+(\mathrm{J2})\) [1211.6170].

The crucial point is that joins are not arbitrary suprema in hom-posets. They are defined only for compatible families, so the operation formalizes gluing along overlaps rather than unrestricted aggregation. A common misconception is that a join restriction category is merely a restriction category whose hom-sets have joins; the compatibility condition and the interaction with composition are essential.

## 2. Compatibility, gluing, and basic examples

The prototypical example is \(\Set_p\), the category of sets and partial functions. Here \(\bar f\) is the partial identity defined on exactly the domain of definition of \(f\). Compatible families of partial functions agree wherever both are defined, and may be glued by union. This makes \(\Set_p\) a join restriction category; in the standard intuition, joins glue partial maps which agree on overlaps [1801.07390].

A second basic example is \(\Top_p\), the category of partial continuous maps of topological spaces. This extends the same partial-map intuition to topology. The examples \(\Set_p\) and \(\Top_p\) show that join restriction structure is not tied to combinatorial partiality: it also captures local geometric data.

The example \(\Rec\), partial recursive functions, is equally important because it marks a boundary. It is not a join restriction category: it lacks joins of arbitrary compatible families unless they are recursively indexed. In the associated would-be enrichment, the presheaf hom-objects fail the sheaf condition for infinite covers; there are matching families of partial recursive functions whose indexing is not recursive, so no unique glue exists in \(\Rec\). The failure is therefore not merely order-theoretic but sheaf-theoretic [1211.6170].

These examples isolate the operative principle. Join restriction structure expresses the existence of global partial maps assembled from local pieces, but only when the ambient category supports genuine amalgamation. This suggests that join restriction categories are best understood as categories of partial maps with an internal gluing discipline, rather than as ordinary categories with extra lattice structure.

## 3. Enriched-categorical formulation

A central result identifies join restriction categories with enriched categories in the sense of Kelly for a suitable enrichment base. The relevant base is a weak double category \(\jR\) whose objects are meet-semilattices \(A\), that is, the objects of
\[
\Gamma(\Stab^{\op})\;\subset\;\twoCat.
\]
Horizontal arrows \(A\rightsquigarrow B\) in \(\jR\) are sheaves
\[
U\;\in\;\Sh\bigl(\Gamma(\Stab^{\op})(A,B)\bigr),
\]
namely functors on the poset of restriction-maps from \(A\) to \(B\) sending covering sieves to limit diagrams. Each such hom-object is a frame, indeed a locale. Vertical arrows \(f\colon A\to A'\) are the morphisms in \(\Gamma(\Stab^{\op})\) that are locally étale, equivalently invertible-on-objects local discrete fibrations whose fibres are sheaves. Horizontal composition is Day convolution restricted to sheaves:
\[
W\otimes U
=
\Lan_{(g,k)}(W(-)\times U(-))\;\longrightarrow\;\Sh(\Gamma(\Stab^{\op})(A,C)),
\]
and the units are the representable presheaves \(\yon_A\). Each hom-category of \(\jR\) is a poset, ordered by inclusion of sheaves [1211.6170].

The main theorem is the 2-equivalence
\[
\jrcat
\;\simeq\;
\jR\text{-}\Cat,
\]
between the 2-category of join restriction categories, join-restriction functors, and total transformations, and the 2-category of categories enriched in \(\jR\). For a join restriction category \(\C\), the enriched category \(\C_{\!en}\) has the same objects, and for \(A,B\in\C\) the hom-object is the sheaf
\[
U_{A,B}\;=\;\bigl(\C(A,B),\,\le\bigr)\;\in\;\Sh\bigl(\Gamma(\Stab^{\op})(A,B)\bigr),
\]
sending each arrow \(f\) to the down-closure \(\{g\mid g\le f\}\). Enriched composition
\[
U_{B,C}\otimes U_{A,B}\to U_{A,C}
\]
is induced by ordinary composition in \(\C\), via the universal property of Day convolution, while identities come from the Yoneda units. Conversely, a \(\jR\)-category recovers a join restriction category whose hom-orders are the frame-orders of the hom-sheaves and whose restriction idempotents arise as unit-component idempotents in the enriched structure [1211.6170].

This characterization is structurally significant because composition and gluing become instances of Day convolution and colimits in frame-valued hom-objects. The same approach extends to range restriction categories by replacing \(\Stab^{\op}\) with the subcategory of open maps \(\oStab^{\op}\), yielding a sub-double-category \(\rR\subset\jR\) and an identification of range restriction categories with \(\rR\)-enriched categories.

## 4. Presheaves, sheaves, and free cocompletion

The presheaf theory of join restriction categories parallels ordinary presheaf theory, but with compatibility and joins built into the data. A restriction presheaf on a restriction category \(\C\) is a functor
\[
P\colon\C^{\mathrm{op}}\longrightarrow\Set
\]
equipped with a restriction idempotent \(\overline x\in\C(A,A)\) for each \(x\in P(A)\), satisfying restriction-presheaf analogues of the axioms. If \(\C\) is a join restriction category, a join restriction presheaf is a restriction presheaf \(P\) such that each \(P(A)\) carries finite or arbitrary joins of compatible elements and these joins satisfy
\[
\overline{\bigvee_{x\in S}x}
\;=\;
\bigvee_{x\in S}\overline x,
\qquad
\Bigl(\bigvee_{x\in S}x\Bigr)\cdot g
\;=\;
\bigvee_{x\in S}(x\cdot g).
\]
The join restriction presheaves on \(\C\) form a join restriction category
\[
\widehat{\C}_j.
\]
Restriction idempotents on natural transformations are computed componentwise, and joins of compatible families of transformations are taken pointwise [1801.07390].

For a small join restriction category \(\C\), the category \(\widehat{\C}_j\) is equivalent to a partial-map category of sheaves. The construction proceeds through the split-completion \(K_r(\C)\), whose total maps form an ordinary category \(\D=\Tot(K_r(\C))\) equipped with a stable system of monics \(M\). One defines a Grothendieck topology \(J\) on \((\D,M)\) whose basic covers are those families of monics whose join in the sup-lattice of subobjects is \(1\). Denoting by \(Sh(\D,J)\) the sheaf topos and by \(\M_\sheaf\) the monomorphisms of sheaves that are locally in \(M\), one obtains
\[
Par\bigl(Sh(\D,J),\,\M_\sheaf\bigr)
\;\simeq\;
\widehat{\C}_j
\]
as join restriction categories [1801.07390].

The conceptual content of this equivalence is explicit: a sheaf matching-family along an \(M\)-cover corresponds exactly to a compatible family in the presheaf, and amalgamation of the matching family corresponds to the join of the compatible family. The sheaf condition is therefore the presheaf-theoretic form of join-gluing.

The Yoneda embedding also persists in join-restriction form. The join restriction Yoneda embedding
\[
y_j\colon\C \;\longrightarrow\;\widehat\C_j,
\qquad
A\;\mapsto\;\C(-,A)
\]
exhibits \(\widehat\C_j\) as the free cocompletion of \(\C\) in the 2-category of join restriction categories. Concretely, for any cocomplete join restriction category \(\E\), precomposition with \(y_j\) induces an equivalence
\[
\bigl[\widehat\C_j,\E\bigr]_{\rm cocont}
\;\simeq\;
\bigl[\C,\E\bigr]_{\rm jrest}.
\]
This places join restriction categories within a cocompletion theory directly analogous to ordinary presheaf completion, but with sheaf-theoretic gluing replacing unrestricted colimit formation [1801.07390].

## 5. Classical structure and distributive restriction categories

Join restriction categories acquire an additional layer of structure in the presence of distributivity and relative complements. A distributive restriction category is a restriction category with finite coproducts \(\oplus\) whose injections are total, a restriction terminal object \(\mathsf1\), restriction products \(\times\), and distributivity isomorphisms
\[
(A\times B)\oplus(A\times C)\;\cong\;A\times(B\oplus C),
\qquad
\mathsf0\;\cong\;A\times\mathsf0.
\]
A classical restriction category is a join restriction category in which for \(f\le g\) there is a relative complement
\[
g\setminus f
\quad:\quad
(g\setminus f)\perp f,
\quad
(g\setminus f)\vee f=g.
\]
Such categories support classical Boolean reasoning [2305.16524].

The key theorem states that in a distributive restriction category,
\[
A\!\widehat{\times}\!B\;:=\;A\oplus B\oplus(A\times B)
\]
with classical projections
\[
p_0\;=\;[\,1_A,\;0,\;\pi_0\,],
\qquad
p_1\;=\;[\,0,\;1_B,\;\pi_1\,]
\]
is an ordinary product of \(A\) and \(B\) if and only if the category is classical. Equivalently, classicality is characterized by the existence of these classical products. In the forward direction, joins and relative complements are recovered from the universal property of this product; in the reverse direction, the required pairing is constructed explicitly using joins and complements [2305.16524].

The same paper also relates this phenomenon to classifiedness and exception monads. A distributive restriction category has a categorical product if and only if it is classified in the sense that every map \(A\to B\) factors uniquely through a total map \(A\to B\oplus\mathsf1\). More specifically,
\[
\text{“classical products and distributive”}
\;\Longleftrightarrow\;
\text{“classically classified”}
\;\Longleftrightarrow\;
\text{Kleisli category of }X\mapsto X\oplus1\text{ on a distributive category.}
\]
Thus the existence of classical products has a strong structural effect: it characterizes precisely the Kleisli situation for the exception monad [2305.16524].

In \(\mathsf{PAR}\), the category of sets and partial functions, joins of compatible partial functions are unions of graphs, relative complements remove the part already covered by a smaller map, and the classical product is
\[
X\oplus Y\oplus (X\times Y).
\]
This reproduces the well-known equivalence
\[
\mathsf{PAR}\;\cong\;\mathsf{SET}_{(\_\,\sqcup1)}.
\]
The example shows that join restriction structure can support not only gluing but also a genuinely classical calculus of partiality [2305.16524].

## 6. Internal étale, tangent, and bundle-theoretic extensions

Join restriction categories also serve as an ambient setting for internal geometric constructions. In a join restriction category \(\mathcal C\), a local homeomorphism \(p\colon X\to Y\) is a total map that can be written as a join of partial isomorphisms. A local atlas on an object \(X\) consists of restriction idempotents
\[
\varphi_{ij}\in(X)
\qquad (i,j\in I)
\]
satisfying symmetry and transitivity, and a glueing of such an atlas is a local homeomorphism
\[
p\;:\;A=\displaystyle\bigvee_{i\in I}p_i\;\longrightarrow\;X
\]
whose basis of partial inverse sections induces exactly the atlas. A join restriction category has local glueings if every local atlas admits a glueing [2004.09699].

Under this hypothesis, there is a broad correspondence between internal categorical structures in \(\mathcal C\) and restriction-theoretic structures over \(\mathcal C\). The paper constructs a Galois adjunction
\[
\Psi \dashv \Phi : \mathcal B \to \mathcal A
\]
between a 2-category of partite internal categories in \(\mathcal C\) and a 2-category of join restriction categories equipped with a restriction functor to \(\mathcal C\). Its fixpoints are, on one side, source-étale partite internal categories, and on the other side, join restriction categories hyperconnected over \(\mathcal C\). Hence one obtains an equivalence between source-étale partite internal categories in \(\mathcal C\) and join restriction categories hyperconnected over \(\mathcal C\). The resulting framework recovers classical correspondences for \(\Top_p\), \(\Loc_p\), and \(\Pos_p\), and provides internal versions of Haefliger groupoids and completion processes [2004.09699].

A further extension equips join restriction categories with tangent structure. A tangent join-restriction category is a join-restriction category \(\mathfrak X\) equipped with a restriction-preserving tangent functor \(T\), additive bundle structure \(p,0,+\), and total maps \(\ell\) and \(c\) satisfying the usual tangent-category axioms in the restriction sense. In \(\mathrm{Man}_p\), the category of manifolds and smooth maps defined on open subsets, the ordinary tangent functor preserves restrictions, so \(\mathrm{Man}_p\) is a Cartesian tangent join-restriction category [2509.18410].

Within such a category, if \(G\) is a group object and the pullback \(T(G)_u\) exists, there is a natural isomorphism
\[
\phi: T(G)\;\xrightarrow{\,\cong\,}\; G\times T(G)_u.
\]
On the identity fibre \(T(G)_u\), the vector-addition and group-multiplication monoid structures coincide by an Eckmann–Hilton argument, yielding a commutative group. Left-invariant vector fields are identified with points of \(T(G)_u\), and the induced bracket makes
\[
\Gamma:=\mathrm{LInv}(G,T(G))\cong T(G)_u
\]
into a Lie algebra over \(\mathbb Z\) [2509.18410].

The same paper formulates fibre bundles and principal bundles entirely in the language of join restriction categories. A fibre bundle with base \(M\) and fibre \(F\) is a total map \(q:E\to M\) together with local trivializations \(\alpha_i:E\to M\times F\) whose domains cover \(E\). A principal \(G\)-bundle is the special case \(F=G\), \(a=m:G\times G\to G\). Every principal \(G\)-bundle \(q:P\to M\) admits a total right action
\[
r:P\times G\;\to\;P
\]
making \(P\) into a torsor in the slice \(\mathfrak X/M\), and, in the presence of gluings, there is an equivalence
\[
\mathrm{Bun}_G(\mathfrak X)\;\simeq\;\mathrm{PBun}_G(\mathfrak X)\;\times\;G\text{-}\mathrm{Obj}(\mathfrak X).
\]
For \(\mathfrak X=\mathrm{Man}_p\) or \(\mathrm{Top}_p\), the totally fibred surjective objects recover the classical notions of smooth and topological principal bundles [2509.18410].

Taken together, these developments indicate that join restriction categories are not merely an abstract formalism for partial functions. They provide a stable categorical language for local-to-global assembly across several domains: enriched hom-objects, sheaf-like cocompletion, Boolean partiality, internal étale structures, and tangent-geometric bundle theory.

Source: https://www.emergentmind.com/topics/join-restriction-categories