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add the category of pseudo-metric spaces
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‎database/data/001_categories/002_analysis.sql‎

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@@ -39,6 +39,16 @@ VALUES
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'https://ncatlab.org/nlab/show/Met',
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NULL
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),
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(
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'PMet',
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'category of pseudo-metric spaces with non-expansive maps',
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'$\mathbf{PMet}$',
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'pseudo-metric spaces',
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'non-expansive maps $f$, meaning $d(f(x),f(y)) \leq d(x,y)$ for all $x,y$',
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'In contrast to metric spaces, we do not demand $d(x,y)=0 \implies x=y$ here.',
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NULL,
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NULL
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),
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(
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'Met_oo',
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'category of metric spaces with ∞ allowed',

‎database/data/001_categories/100_related-categories.sql‎

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@@ -66,6 +66,7 @@ VALUES
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('Met', 'Met_c'),
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('Met', 'Met_oo'),
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('Met', 'Ban'),
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('Met', 'PMet'),
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('Met_c', 'Met'),
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('Met_c', 'Met_oo'),
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('Met_c', 'Top'),
@@ -80,6 +81,7 @@ VALUES
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('N_oo', 'N'),
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('N_oo', 'On'),
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('On', 'N'),
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('PMet', 'Met'),
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('Pos', 'FinOrd'),
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('Pos', 'Prost'),
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('Prost', 'Pos'),

‎database/data/002_tags/002_category-tags.sql‎

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('N_oo', 'thin'),
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('On', 'set theory'),
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('On', 'thin'),
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('PMet', 'analysis'),
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('Pos', 'order theory'),
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('Prost', 'order theory'),
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('R-Mod', 'algebra'),

‎database/data/004_property-assignments/PMet.sql‎

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INSERT INTO category_property_assignments (
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category_id,
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property_id,
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is_satisfied,
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reason
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)
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VALUES
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(
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'PMet',
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'locally small',
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TRUE,
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'There is a forgetful functor $\mathbf{PMet} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
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),
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(
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'PMet',
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'equalizers',
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TRUE,
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'Just restrict the pseudo-metric to the equalizer built from the sets.'
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),
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(
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'PMet',
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'binary products',
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TRUE,
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'The product of two pseudo-metric spaces $(X,d)$, $(Y,d)$ is $(X \times Y,d)$ with $d((x_1,y_1),(x_2,x_2)) := \sup(d(x_1,x_2),d(y_1,y_2))$.'
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),
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(
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'PMet',
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'terminal object',
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TRUE,
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'The one-point (pseudo-)metric space is terminal.'
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),
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(
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'PMet',
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'coequalizers',
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TRUE,
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'See <a href="https://mathoverflow.net/questions/123739" target="_blank">MO/123739</a>.'
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),
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(
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'PMet',
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'directed colimits',
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TRUE,
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'Given a directed diagram $(X_i)$ of pseudo-metric spaces, take the directed colimit $X$ of the underlying sets with the following pseudo-metric: If $x,y \in X$, let $d(x,y)$ be infimum of all $d(x_i,y_i)$, where $x_i,y_i \in X_i$ are some preimages of $x,y$ in some $X_i$. The definition ensures that each $X_i \to X$ is non-expansive, and the universal property is easy to check.'
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),
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(
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'PMet',
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'exact filtered colimits',
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TRUE,
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'Let $\mathcal{I}$ be a finite category and $\mathcal{J}$ be a small filtered category, w.l.o.g. a directed poset. Let $X : \mathcal{I} \times \mathcal{J} \to \mathbf{PMet}$ be a diagram. We need to show that the canonical map $\mathrm{colim}_{j \in \mathcal{J}} \lim_{i \in \mathcal{I}} X(i,j) \to \lim_{i \in \mathcal{I}} \mathrm{colim}_{j \in \mathcal{J}} X(i,j)$ is an isomorphism. It is bijective since the forgetful functor to $\mathbf{Set}$ preserves finite limits and filtered colimits and since $\mathbf{Set}$ has exact filtered colimits. That the map is isometric can easily be reduced to the following lemma: If $d_{i,j} \in \mathbb{R}_{\geq 0}$ are numbers for $i \in \mathcal{I}$, $j \in \mathcal{J}$ with $j \leq k \implies d_{i,k} \leq d_{i,j}$, then $\inf_j \sup_i d_{i,j} = \sup_i \inf_j d_{i,j}$. This can be proven directly. Alternatively, use that the thin category $(\mathbb{R}_{\geq 0} \cup \{\infty\},\leq)$ is isomorphic to $([0,1],\leq)$, and we already know that <a href="/category/real_interval">it has exact filtered colimits</a>.'
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),
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(
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'PMet',
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'strict initial object',
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TRUE,
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'The empty (pseudo-)metric space is initial and clearly strict.'
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),
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(
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'PMet',
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'generator',
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TRUE,
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'The one-point (pseudo-)metric space is a generator since it represents the forgetful functor $\mathbf{PMet} \to \mathbf{Set}$.'
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),
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(
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'PMet',
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'cogenerator',
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TRUE,
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'The set $\{0,1\}$ equipped with the pseudo-metric $d(0,1)=0$ is a cogenerator since every map into is automatically non-expansive and since $\{0,1\}$ is a cogenerator in $\mathbf{Set}$.'
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),
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(
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'PMet',
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'semi-strongly connected',
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TRUE,
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'Every non-empty pseudo-metric space is weakly terminal (by using constant maps).'
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),
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(
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'PMet',
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'well-powered',
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TRUE,
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'This follows since monomorphisms are injective (see below).'
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),
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(
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'PMet',
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'well-copowered',
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TRUE,
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'This follows since epimorphisms are surjective (see below).'
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),
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(
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'PMet',
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'balanced',
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FALSE,
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'Let $d : \mathbb{R} \times \mathbb{R} \to \mathbb{R}_{\geq 0}$ be the usual Euclidean metric on $\mathbb{R}$ and $0 : \mathbb{R} \times \mathbb{R} \to \mathbb{R}_{\geq 0}$ be the zero pseudo-metric. Then the identity map $(\mathbb{R},d) \to (\mathbb{R},0)$ provides a counterexample.'
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),
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(
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'PMet',
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'cartesian closed',
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FALSE,
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'This is proven in <a href="https://math.stackexchange.com/questions/5131457" target="_blank">MSE/5131457</a>.'
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),
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(
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'PMet',
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'countable powers',
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FALSE,
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'Assume that the power $P = \mathbb{R}^{\mathbb{N}}$ exists, where $\mathbb{R}$ has the usual (pseudo-)metric. Since the forgetful functor $\mathbf{PMet} \to \mathbf{Set}$ is representable, it preserves limits, powers in particular. Thus, we may assume that $P$ is the set of sequences of numbers and that the projections $p_n : P \to \mathbb{R}$ are given by $p_n(x) = x_n$. Now consider the sequences $x = (n)_n$ and $y = (0)_n$. Since each $p_n$ is non-expansive, we get $d(x,y) \geq d(p_n(x),p_n(y)) = d(n,0) = n$. But then $d(x,y) = \infty$, a contradiction.'
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),
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(
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'PMet',
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'binary copowers',
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FALSE,
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'The coproduct of two non-empty pseudo-metric spaces does not exist, see <a href="https://math.stackexchange.com/questions/1778408" target="_blank">MSE/1778408</a> (the proof also works for pseudo-metric spaces). For example, the copower $\mathbb{R} \sqcup \mathbb{R}$ does not exist. We only get coproducts when allowing $\infty$ as a distance.'
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),
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(
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'PMet',
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'strict terminal object',
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FALSE,
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'This is trivial.'
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),
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(
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'PMet',
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'essentially small',
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FALSE,
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'This is trivial.'
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),
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(
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'PMet',
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'skeletal',
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FALSE,
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'This is trivial.'
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),
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(
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'PMet',
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'Malcev',
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FALSE,
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'Take any counterexample in $\mathbf{Set}$ and equip it with the zero pseudo-metric.'
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),
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(
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'PMet',
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'natural numbers object',
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FALSE,
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'If $(N,z,s)$ is a natural numbers object in $\mathbf{PMet}$, then
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<p>$1 \xrightarrow{z} N \xleftarrow{s} N$</p>
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is a coproduct cocone by <a href="https://ncatlab.org/nlab/show/Sketches+of+an+Elephant" target="_blank">Johnstone</a>, Part A, Lemma 2.5.5. Since there is a map $1 \to N$, we have $N \neq \varnothing$. However, the coproduct of two non-empty pseudo-metric spaces does not exist, see <a href="https://math.stackexchange.com/questions/1778408" target="_blank">MSE/1778408</a>.'
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);

‎database/data/006_special-objects/002_initial_objects.sql‎

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('N', '$0$'),
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('N_oo', '$0$'),
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('On', '$0$'),
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('PMet', 'empty pseudo-metric space'),
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('Pos', 'empty poset'),
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('Prost', 'empty proset'),
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('R-Mod', 'trivial module'),

‎database/data/006_special-objects/003_terminal_objects.sql‎

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('Met_oo', 'singleton space'),
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('Mon', 'trivial monoid'),
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('N_oo', '$\infty$'),
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('PMet', 'singleton space'),
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('Pos', 'singleton poset'),
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('Prost', 'singleton proset'),
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('R-Mod', 'zero module'),

‎database/data/006_special-objects/005_products.sql‎

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('FreeAb', '[finite case] direct sums'),
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('Met_c', '[countable case] In the finite case, take direct products with the metric $d(x,y) = \sup_i d_i(x_i,y_i)$, but other metrics such as $d(x,y) = \sum_i d_i(x_i,y_i)$ also work. In the countable case, one can assume $d_i \leq 1$ and then define $d(x,y) = \sum_i d_i(x,y) / 2^i$.'),
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('Met', '[finite case] direct products with the metric $d(x,y) = \sup_i d_i(x_i,y_i)$'),
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('PMet', '[finite case] direct products with the pseudo-metric $d(x,y) = \sup_i d_i(x_i,y_i)$'),
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('Sch', '[finite case] The idea is to use $\mathrm{Spec}(A) \times \mathrm{Spec}(B) = \mathrm{Spec}(A \otimes B)$ and then to glue affine pieces together. See EGA I, Chap. I, Thm. 3.2.1.'),
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('Man', '[finite case] direct products $X \times Y$ with the product topology and the charts $\mathbb{R}^{n + m} = \mathbb{R}^n \times \mathbb{R}^m \cong U \times V \hookrightarrow X \times Y$ for charts $\mathbb{R}^n \cong U \hookrightarrow X$ and $\mathbb{R}^m \cong V \hookrightarrow Y$'),
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('walking_span', '[binary case] $1 \times 2 = 0$, $x \times x = x$, $0 \times x = 0$');

‎database/data/007_special-morphisms/002_isomorphisms.sql‎

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'only the identities',
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'This is true for every poset (regarded as a category).'
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),
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(
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'PMet',
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'bijective isometries',
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'This is easy.'
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),
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(
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'Pos',
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'bijective functions that are order-preserving and order-reflecting',

‎database/data/007_special-morphisms/003_monomorphisms.sql‎

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'every morphism',
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'It is a thin category.'
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),
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(
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'PMet',
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'injective non-expansive maps',
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'For the non-trivial direction, the forgetful functor to $\mathbf{Set}$ is representable (by the terminal object), hence preserves monomorphisms.'
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),
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(
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'Pos',
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'injective order-preserving functions',

‎database/data/007_special-morphisms/004_epimorphisms.sql‎

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'every morphism',
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'It is a thin category.'
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),
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(
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'PMet',
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'surjective non-expansive maps',
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'Let $f : X \to Y$ be a non-expansive map that is not surjective. Choose $y_0 \in Y \setminus f(X)$. We extend the pseudo-metric from $Y$ to $Z := Y \sqcup \{y''_0\}$ via $d(y,y''_0) := d(y,y_0)$, i.e., we make $y_0,y''_0$ indistinguishable. Let $g : Y \to Z$ be the inclusion and $h : Y \to Z$ be the map that composes $g$ with the swap between $y_0$ and $y''_0$. Both are isometric and satisfy $g \circ f = h \circ f$. Therefore, $f$ is not an epimorphism.'
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),
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(
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'On',
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'every morphism',

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