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A database of categories


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Revision2f7600df8944b45de89d0d2aaddfe851e6407fde (tree)
Time2021-09-23 03:54:38
AuthorCorbin <cds@corb...>
CommiterCorbin

Log Message

Clean up opposite and dagger categories.

Change Summary

Incremental Difference

Binary files a/catabase.db and b/catabase.db differ
--- /dev/null
+++ b/templates/row-catabase-dagger_categories.html
@@ -0,0 +1,31 @@
1+{% extends "default:row.html" %}
2+
3+{% block content %}
4+
5+{% set cat = display_rows[0]["category"] %}
6+
7+<h1>†-category: {{ cat }}</h1>
8+
9+<p>{{ cat }} is a †-category. Composition in {{ cat }} can happen in
10+either direction; every arrow can interchange its source and target objects
11+freely.</p>
12+
13+{% set core_rows = sql("select groupoid from cores where category = ?", [cat]) %}
14+{% if core_rows %}
15+{% set grpd = core_rows[0]["groupoid"] %}
16+{% set unitary_rows = sql(
17+"select arrows_hr from categories where name = ?", [grpd]) %}
18+{% set unitary_arrows = unitary_rows[0]["arrows_hr"] %}
19+<p>The core of {{ cat }} is the groupoid {{ grpd }}:</p>
20+
21+<div class="bigmath">
22+ Core({{ cat }}) ≅ {{ grpd }}
23+</div>
24+
25+<p>The unitary arrows of {{ cat }} are the arrows of {{ grpd }}; they are
26+{{ unitary_arrows }}.</p>
27+{% endif %}
28+
29+{{ super() }}
30+{% endblock %}
31+
--- a/templates/row-catabase-opposite_categories.html
+++ b/templates/row-catabase-opposite_categories.html
@@ -12,9 +12,10 @@
1212 {% else %}
1313 <h1>Opposite Categories: {{ cat }} &amp; {{ opcat }}</h1>
1414
15-<p>{{ cat }} and {{ opcat }} are opposites; they are equivalent, except that
16-the arrows in {{ cat }} are pointed in the opposite direction from in
17-{{ opcat }}.</p>
15+<p>{{ cat }} and {{ opcat }} are opposites; they share the same objects,
16+identity arrows, and composition, but the arrows in {{ cat }} are pointed in
17+the opposite direction from in {{ opcat }}. This is a contravariant
18+equivalence of categories.</p>
1819 {% endif %}
1920
2021 {{ super() }}
--- /dev/null
+++ b/templates/table-catabase-dagger_categories.html
@@ -0,0 +1,39 @@
1+{% extends "default:table.html" %}
2+
3+{% block content %}
4+
5+<h1>†-categories</h1>
6+
7+<p>A †-category ("dagger category") is like a category where composition can
8+happen in either direction. More precisely, the arrows of a †-category can
9+freely interchange their source and target objects. Intuitively, composition
10+in a category must follow a directed path, but composition in a †-category is
11+undirected.</p>
12+
13+<p>A †-functor ("dagger functor") is like a functor, but with an additional
14+rule for commuting everything. There is a 2-category DagCat of †-categories,
15+†-functors, and natural transformations.</p>
16+
17+<p>There is no canonical forgetful 2-functor from †-categories to
18+categories. For any particular choice of arrow direction, there is a
19+2-functor U which sends each †-category to a category:</p>
20+
21+<div class="bigmath">
22+ U : DagCat → Cat
23+</div>
24+
25+<p>And equips that category with an involutive endofunctor † which is the
26+identity on objects:</p>
27+
28+<div class="bigmath">
29+ † : X ↦ X
30+</div>
31+
32+<p>While U might not be canonical, it will always choose the same categories
33+and involutive endofunctors up to equivalence, so we may speak of {{ cat }} as
34+a category without ambiguity.</p>
35+
36+<p>The arrows of the core of a †-category are called unitary arrows.</p>
37+
38+{{ super() }}
39+{% endblock %}
--- a/templates/table-catabase-restriction_categories.html
+++ b/templates/table-catabase-restriction_categories.html
@@ -1,4 +1,4 @@
1-{% extends "default:row.html" %}
1+{% extends "default:table.html" %}
22
33 {% block content %}
44
--- /dev/null
+++ b/todo.txt
@@ -0,0 +1,46 @@
1+* Catabase:
2+ * all_ polymorphism
3+ * All skeletal subcategories are equivalences of categories;
4+ maybe `all_equivalent_categories`?
5+ * Would handle opposite categories too...
6+ * `all_groupoids`? Would include cores and homotopy categories?
7+ * Random facts not yet encodable
8+ * P and FinSet are symmetric rig categories
9+ * So is Vect_k, but not with categorical product
10+ * MonCat has a double category, using "oplax" monoidal functors
11+ * Monoidal categories in general
12+ * Needs polymorphism
13+ * Categorical product -> Cartesian closed
14+ * Many other cases to handle
15+ * The Euler characteristic AKA groupoid cardinality of P is Euler's
16+ constant e ~ 2.718
17+ * Functor categories: structure types, ...
18+ * Presheaf categories: FinSet, Species, ...
19+ * The span construction: Span(Set) and Span(Grpd) are 2-categories!
20+ * The homotopy-category construction: Ho(Cat), Ho(Top), ...
21+ * Arrow categories: Sierpinski topos, ...
22+ * Topoi <-> categories of sheaves on spaces
23+ * CCCs: DagCat, ...
24+ * Dismantle `enrichments`
25+ * Give 2-categories their own table and `natural_transformations_hr` for
26+ a row template page
27+ * Manage Evil
28+ * 2-categories are more Evil than bicategories
29+ * Double categories should be weak by default
30+ * Dagger categories should be equipped with structure
31+ * Upstream work
32+ * Indexing for categories: Field, nCob, Vect_k, Mat_R, Mod(Ab)
33+ * Field and nCob are disconnected by index!
34+* Complexity:
35+ * Refactor `is_contained_in` and `is_not_contained_in` to views
36+ * Imply containment and separation using example problems from
37+ `is_element_of` and `is_not_element_of`
38+* nLab:
39+ * Separation axioms: Add zaha-style diagrams?
40+ * Shutt expressiveness: Create
41+ * Shutt abstraction: Create
42+ * meros: Create
43+ * Theorems for Free!: This should be on nLab, shouldn't it?
44+ * Haskell: Add links
45+ https://www.cs.ox.ac.uk/jeremy.gibbons/publications/fast+loose.pdf and
46+ https://wiki.haskell.org/Hask to justify why Hask is not categorical