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A database of categories


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Revision5f48895d0069b686eb6a43605ee117304d7b874d (tree)
Time2021-09-11 12:26:18
AuthorCorbin <cds@corb...>
CommiterCorbin

Log Message

Refactor more subcategories.

Add double categories, fix up incorrect parts of simplicial objects,
improve restriction categories.

Change Summary

Incremental Difference

Binary files a/catabase.db and b/catabase.db differ
--- a/templates/row-catabase-categories.html
+++ b/templates/row-catabase-categories.html
@@ -51,28 +51,17 @@ SSet).</p>
5151 <p>{{ name }} is semiadditive; products and sums coincide (enriched in CMon).</p>
5252 {% endif %}
5353
54-{% set subcats = sql("select subcategory from subcategories where parent = ?", [name]) %}
54+{% set subcats = sql(
55+"select * from all_subcategories where supercategory = ?", [name]) %}
5556 {% if subcats %}
56-<h2>Subcategories</h2>
57+<h2>Notable Subcategories</h2>
5758 <ul>
5859 {% for subcat in subcats %}
59- <li>{{ subcat["subcategory"] }}</li>
60+ <li>{{ subcat["subcategory"] }} ({{ subcat["kind"] }})</li>
6061 {% endfor %}
6162 </ul>
62-{% endif %}
63-
64-<h2>Derived Categories</h2>
65-{% set birel = sql("select internal_relations from bicategories_of_relations where parent = ?", [name]) %}
66-{% if birel %}
67-<p>The bicategory-of-relations construction yields the category {{ birel[0]["internal_relations"] }}</p>
68-{% endif %}
69-{% set mon = sql("select * from categories_of_monoids where parent = ?", [name]) %}
70-{% if mon %}
71-<p>The internal monoids form the category {{ mon[0]["internal_monoids"] }}</p>
72-{% endif %}
73-{% set cmon = sql("select * from categories_of_commutative_monoids where parent = ?", [name]) %}
74-{% if cmon %}
75-<p>The internal commutative monoids form the category {{ cmon[0]["internal_commutative_monoids"] }}</p>
63+{% else %}
64+<p>({{ name }} doesn't seem to have any notable subcategories.)</p>
7665 {% endif %}
7766
7867 <h2>Limits</h2>
--- a/templates/row-catabase-categories_of_simplicial_objects.html
+++ b/templates/row-catabase-categories_of_simplicial_objects.html
@@ -7,14 +7,14 @@
77
88 <h1>Category of Simplicial Objects: {{ sscat }}</h1>
99
10-<p>{{ sscat }} is a full subcategory of {{ cat }} whose objects are all
11-internal simplicial objects. A simplicial object in {{ cat }} is a
12-contravariant functor from Δ to {{ cat }}, so the category of simplicial
13-objects is a contravariant functor category:</p>
10+<p>{{ sscat }} is a contravariant functor category from Δ to {{ cat }}:</p>
1411
1512 <div class="bigmath">
1613 {{ sscat }} ≅ [Δ°, {{ cat }}]
1714 </div>
1815
16+<p>The objects of {{ sscat }} are traditionally called "simplicial objects",
17+and {{ sscat }} is a category of simplicial objects.</p>
18+
1919 {{ super() }}
2020 {% endblock %}
--- /dev/null
+++ b/templates/row-catabase-double_categories.html
@@ -0,0 +1,39 @@
1+{% extends "default:row.html" %}
2+
3+{% block content %}
4+
5+{% set v2cat = display_rows[0]["vertical_2cat"] %}
6+{% set h2cat = display_rows[0]["horizontal_2cat"] %}
7+{% set squares = display_rows[0]["2cells_hr"] %}
8+{% set vrow = sql(
9+"select objects_hr, arrows_hr from categories where name = ?", [v2cat])[0] %}
10+{% set hrow = sql(
11+"select objects_hr, arrows_hr from categories where name = ?", [h2cat])[0] %}
12+{% set vobjs = vrow["objects_hr"] %}
13+{% set varrs = vrow["arrows_hr"] %}
14+{% set hobjs = hrow["objects_hr"] %}
15+{% set harrs = hrow["arrows_hr"] %}
16+
17+{% set objs = vobjs if vobjs == hobjs else vobjs + " or " + hobjs %}
18+
19+<h1>Double Category: <u>{{ v2cat }}</u></h1>
20+
21+<p><u>{{ v2cat }}</u> is a double category assembled from the categories
22+{{ v2cat }} and {{ h2cat }}. Its objects are {{ objs }}, its horizontal
23+arrows are {{ harrs }}, its vertical arrows are {{ varrs }}, and its
24+squares are {{ squares }}. As with all double categories, we may transpose
25+horizontal and vertical arrows to obtain another double category.</p>
26+
27+<p>The horizontal and vertical 2-category functors send <u>{{ v2cat }}</u> to
28+the 2-categories {{ h2cat }} and {{ v2cat }} respectively, augmented with
29+{{ squares }} for 2-cells:</p>
30+
31+<div class="bigmath">
32+ 𝓗(<u>{{ v2cat }}</u>) ≅ {{ h2cat }}
33+ <br />
34+ 𝓥(<u>{{ v2cat }}</u>) ≅ {{ v2cat }}
35+</div>
36+
37+{{ super() }}
38+{% endblock %}
39+
--- a/templates/row-catabase-full_subcategories.html
+++ b/templates/row-catabase-full_subcategories.html
@@ -12,5 +12,8 @@
1212 {{ sub }} has only some of the objects of {{ sup }}, but all of the
1313 arrows.</p>
1414
15+<p>In terms of stuff, structure, and properties, objects of {{ sub }} are like
16+objects of {{ sup }} but with extra properties.</p>
17+
1518 {{ super() }}
1619 {% endblock %}
--- /dev/null
+++ b/templates/row-catabase-reflective_subcategories.html
@@ -0,0 +1,35 @@
1+{% extends "default:row.html" %}
2+
3+{% block content %}
4+
5+{% set subcat = display_rows[0]["subcategory"] %}
6+{% set supercat = display_rows[0]["supercategory"] %}
7+{% set reflector = display_rows[0]["reflector_hr"] %}
8+{% set reflection = display_rows[0]["reflection_hr"] %}
9+
10+<h1>Reflective Subcategory: {{ subcat }}</h1>
11+
12+<p>{{ subcat }} is a reflective subcategory of {{ supercat }}, by which we
13+mean that {{ subcat }} is a full subcategory whose inclusion functor has
14+a left adjoint. The inclusion functor i is called the <dfn>reflection</dfn>
15+and the left adjoint is called the <dfn>reflector</dfn>.</p>
16+
17+{% if reflector %}
18+<p>Reflectors tend to be meaningful, and in this case, the functor which
19+reflects {{ supercat }} to {{ subcat }} is called the
20+<em>{{ reflector }} functor</em> of {{ supercat }}.</p>
21+{% endif %}
22+
23+{% if reflection %}
24+<p>Similarly, reflections tend to be meaningful too, and in this case, the
25+functor which includes {{ subcat }} into {{ supercat }} is called the
26+<em>{{ reflection }} functor</em> of {{ subcat }}.</p>
27+{% endif %}
28+
29+<p>In terms of stuff, structure, and properties, objects of {{ supercat }} are like
30+objects of {{ subcat }} but with extra stuff; this is the reflection of the
31+fact that, because {{ subcat }} is a full subcategory, objects of {{ subcat }}
32+are like objects of {{ supercat }} but with extra properties.</p>
33+
34+{{ super() }}
35+{% endblock %}
--- a/templates/row-catabase-restriction_categories.html
+++ b/templates/row-catabase-restriction_categories.html
@@ -8,16 +8,9 @@
88 <h1>Restriction Categories: {{ parcat }}</h1>
99
1010 <p>{{ parcat }} is a restriction category; its arrows are partial maps and it
11-contains {{ subcat }} as a subcategory with total maps.</p>
12-
13-<p>There is a functor from restriction categories to categories:</p>
14-
15-<div class="bigmath">
16- Total : RCat → Cat
17-</div>
18-
19-<p>There are many subcategories, but only {{ subcat }} is the total
20-subcategory of {{ parcat }}:</p>
11+contains {{ subcat }} as a subcategory with total maps. There are many
12+subcategories, but only {{ subcat }} is the total subcategory of
13+{{ parcat }}:</p>
2114
2215 <div class="bigmath">
2316 Total({{ parcat }}) ≅ {{ subcat }}
--- /dev/null
+++ b/templates/table-catabase-all_double_categories.html
@@ -0,0 +1,38 @@
1+{% extends "default:table.html" %}
2+
3+{% block content %}
4+
5+<h1>Double Categories</h1>
6+
7+<p>A double category is a generalized 2-category. A 2-category has objects,
8+arrows, and natural transformations; arrows have one type of composition, but
9+natural transformations have two types of composition, vertical and horizontal
10+composition. A double category has two types of arrows, vertical and
11+horizontal arrows, as well as natural transformations.</p>
12+
13+<p>Every 2-category can be regarded as a double category with only one type of
14+arrow. Similarly, every category can be regarded as a double category with
15+only one type of arrow, using commuting squares of arrows for natural
16+transformations. Every category's slice and coslice categories can be paired
17+to form a double category.</p>
18+
19+<p>By the Macrocosm Principle, there is a category DblCat whose objects are
20+double categories and arrows are double functors; this is the category of
21+internal categories in Cat. There are two functors which designate the
22+2-categories resulting from restricting the vertical and horizontal arrows
23+respectively to identities; they are known as the horizontal and vertical
24+2-categories.</p>
25+
26+<div class="bigmath">
27+ 𝓗, 𝓥 : DblCat → 2Cat
28+</div>
29+
30+<p>There are several common constructions for double categories which are
31+included from other tables:</p>
32+
33+<ul>
34+ <li>Every restriction category gives a double category.</li>
35+</ul>
36+
37+{{ super() }}
38+{% endblock %}
--- a/templates/table-catabase-all_full_subcategories.html
+++ b/templates/table-catabase-all_full_subcategories.html
@@ -8,7 +8,7 @@
88 There are several special cases:</p>
99
1010 <ul>
11- <li>Categories of simplicial objects are full.</li>
11+ <li>Reflective subcategories are full.</li>
1212 </ul>
1313
1414 {{ super() }}
--- /dev/null
+++ b/templates/table-catabase-restriction_categories.html
@@ -0,0 +1,29 @@
1+{% extends "default:row.html" %}
2+
3+{% block content %}
4+
5+<h1>Restriction Categories</h1>
6+
7+<p>A restriction category is a category with a restriction operator which
8+restricts each arrow to a subdomain. By analogy, the original category's
9+arrows are partial, but the arrows of the subcategory obtained by restricting
10+every arrow are total.</p>
11+
12+<p>The restriction operator extends to a functor from restriction categories
13+to categories:</p>
14+
15+<div class="bigmath">
16+ Total : RCat → Cat
17+</div>
18+
19+<p>Additionally, every restriction category forms a double category whose
20+vertical arrows are the original partial arrows and horizontal arrows are the
21+restricted total arrows. This construction also extends to a functor from
22+restriction categories to double categories:</p>
23+
24+<div class="bigmath">
25+ Dc : RCat → DblCat
26+</div>
27+
28+{{ super() }}
29+{% endblock %}