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A database of categories


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Revision8a15f5187e74d3534ae11144f268dfc0ae7f2e3e (tree)
Time2021-09-24 03:29:48
AuthorCorbin <cds@corb...>
CommiterCorbin

Log Message

Factor 2-categories to their own table, and format their row.

Change Summary

Incremental Difference

Binary files a/catabase.db and b/catabase.db differ
--- /dev/null
+++ b/templates/row-catabase-2-categories-0144a1.html
@@ -0,0 +1,25 @@
1+{% extends "default:row.html" %}
2+
3+{% block content %}
4+
5+{% set name = display_rows[0]["name"] %}
6+{% set cat_rows = sql("select * from categories where name = ?", [name]) %}
7+{% set objects = cat_rows[0]["objects_hr"] %}
8+{% set arrows = cat_rows[0]["arrows_hr"] %}
9+{% set transforms = display_rows[0]["transformations_hr"] %}
10+{% set strict = display_rows[0]["is_strict"] %}
11+
12+<h1>2-Category: {{ name }}</h1>
13+
14+<p>{{ name }} is the 2-category whose objects are {{ objects }}, arrows are
15+{{ arrows }}, and transformations are {{ transforms }}. Specifically,
16+{{ name }} is a
17+{% if strict %}
18+strict 2-category.
19+{% else %}
20+bicategory.
21+{% endif %}
22+</p>
23+
24+{{ super() }}
25+{% endblock %}
--- a/templates/row-catabase-categories.html
+++ b/templates/row-catabase-categories.html
@@ -22,9 +22,7 @@
2222 {% if sql("select 1 from is_locally_small where name = ?", [name]) %}
2323 <p>{{ name }} is locally small (enriched in Set).</p>
2424 {% endif %}
25-{% if sql("select 1 from enrichments where category = ? and homs = 'Cat'", [name]) %}
26-<p>{{ name }} is a 2-category (enriched in Cat).</p>
27-{% elif sql("select 1 from enrichments where category = ? and homs = 'Pos'", [name]) %}
25+{% if sql("select 1 from enrichments where category = ? and homs = 'Pos'", [name]) %}
2826 <p>{{ name }} is a 2-poset, a locally posetal or 2-thin 2-category (enriched
2927 in Pos).</p>
3028 {% endif %}
--- a/templates/table-catabase-dagger_categories.html
+++ b/templates/table-catabase-dagger_categories.html
@@ -5,18 +5,18 @@
55 <h1>†-categories</h1>
66
77 <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>
8+happen in either direction. More precisely, the arrows of a category are
9+ordered pairs of objects, but the arrows of a †-category are unordered pairs.
10+Intuitively, composition in a category must follow a directed path, but
11+composition in a †-category creates undirected paths.</p>
12+
13+<p>A †-functor ("dagger functor") is like a functor, but for arrows of
14+†-categories. There is a 2-category DagCat of †-categories, †-functors, and
15+natural transformations.</p>
16+
17+<p>There is no canonical forgetful 2-functor from †-categories to categories,
18+but there is a forgetful functor U which sends each †-category to a
19+category:</p>
2020
2121 <div class="bigmath">
2222 U : DagCat → Cat
--- a/todo.txt
+++ b/todo.txt
@@ -16,17 +16,15 @@
1616 constant e ~ 2.718
1717 * Functor categories: structure types, ...
1818 * Presheaf categories: FinSet, Species, ...
19- * The span construction: Span(Set) and Span(Grpd) are 2-categories!
19+ * The span construction: Span(Set) and Span(Grpd) are 2-categories because
20+ Set and Grpd have pullbacks
2021 * The homotopy-category construction: Ho(Cat), Ho(Top), ...
2122 * Arrow categories: Sierpinski topos, ...
2223 * Topoi <-> categories of sheaves on spaces
2324 * CCCs: DagCat, ...
2425 * Dismantle `enrichments`
2526 * Already banned: Set, Cat
26- * Give 2-categories their own table and `natural_transformations_hr` for
27- a row template page
2827 * Manage Evil
29- * 2-categories are more Evil than bicategories
3028 * Double categories should be weak by default
3129 * Upstream work
3230 * Indexing for categories: Field, nCob, Vect_k, Mat_R, Mod(Ab)
@@ -40,6 +38,7 @@
4038 * Shutt expressiveness: Create
4139 * Shutt abstraction: Create
4240 * meros: Create
41+ * categorical combinators: Create
4342 * Theorems for Free!: This should be on nLab, shouldn't it?
4443 * Haskell: Add links
4544 https://www.cs.ox.ac.uk/jeremy.gibbons/publications/fast+loose.pdf and