-
-
Notifications
You must be signed in to change notification settings - Fork 7
Expand file tree
/
Copy path008_locally-presentable.sql
More file actions
166 lines (166 loc) · 8.49 KB
/
Copy path008_locally-presentable.sql
File metadata and controls
166 lines (166 loc) · 8.49 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
INSERT INTO properties (
id,
relation,
description,
nlab_link,
dual_property_id,
invariant_under_equivalences
)
VALUES
(
'locally finitely presentable',
'is',
'A category is <i>locally finitely presentable</i> if it satisfies one of the following equivalent conditions:
<ol>
<li>It is finitely accessible and cocomplete.</li>
<li>It is finitely accessible and complete.</li>
<li>It is equivalent to the category of finite-limit-preserving functors to $\mathbf{Set}$ from a small category with finite limits.</li>
<li>It is equivalent to the category of models of a small finite-limit sketch.</li>
</ol>
For equivalence of conditions above, see Cor. 2.47, Thm. 1.46, and Cor. 1.52 in <a href="https://ncatlab.org/nlab/show/Locally+Presentable+and+Accessible+Categories" target="_blank">Adamek-Rosicky</a>. This is the same as being locally $\aleph_0$-presentable.',
'https://ncatlab.org/nlab/show/locally+finitely+presentable+category',
NULL,
TRUE
),
(
'locally presentable',
'is',
'Let $\kappa$ be a regular cardinal. A category is <i>locally $\kappa$-presentable</i> if it satisfies one of the following equivalent conditions:
<ol>
<li>It is $\kappa$-accessible and cocomplete.</li>
<li>It is $\kappa$-accessible and complete.</li>
<li>It is equivalent to the category of $\kappa$-limit-preserving functors to $\mathbf{Set}$ from a small category with $\kappa$-limits.</li>
<li>It is equivalent to the category of models of a small $\kappa$-limit sketch.</li>
</ol>
For equivalence of conditions above, see Cor. 2.47, Thm. 1.46, and Cor. 1.52 in <a href="https://ncatlab.org/nlab/show/Locally+Presentable+and+Accessible+Categories" target="_blank">Adamek-Rosicky</a>.
A category is <i>locally presentable</i> if it is locally $\kappa$-presentable for some regular cardinal $\kappa$.',
'https://ncatlab.org/nlab/show/locally+presentable+category',
'locally copresentable',
TRUE
),
(
'locally copresentable',
'is',
'A category is <i>locally copresentable</i> if its opposite category is locally presentable.',
NULL,
'locally presentable',
TRUE
),
(
'locally ℵ₁-presentable',
'is',
'This is the special case of the notion of locally $\kappa$-presentable categories, where $\kappa = \aleph_1$ is the first uncountable cardinal.',
'https://ncatlab.org/nlab/show/locally+presentable+category',
NULL,
TRUE
),
(
'locally strongly finitely presentable',
'is',
'A category is <i>locally strongly finitely presentable</i> if it is cocomplete and there is a set $G$ of strongly finitely presentable objects such that every object is a sifted colimit of objects from $G$.
There are several equivalent conditions:
<ol>
<li>It is equivalent to the category of models of a many-sorted finitary algebraic theory.</li>
<li>It is equivalent to the category of finite-product-preserving functors to $\mathbf{Set}$ from a small category with finite products (=Lawvere theory).</li>
<li>It is equivalent to the category of models of a small finite-product sketch.</li>
<li>It is equivalent to the Eilenberg–Moore category of a finitary (=filtered-colimit-preserving) monad on $\mathbf{Set}^S$ for some set $S$.</li>
<li>It is equivalent to the Eilenberg–Moore category of a sifted-colimit-preserving monad on $\mathbf{Set}^S$ for some set $S$. (cf. [<a href="https://doi.org/10.2168/LMCS-8(3:14)2012" target="_blank">KR12</a>, Proposition 3.3])</li>
</ol>
A category satisfying this property is simply called a <i>variety</i> (of algebras) by some authors, although one should be aware that this term is sometimes used only for the one-sorted case.',
'https://ncatlab.org/nlab/show/locally+strongly+finitely+presentable+category',
NULL,
TRUE
),
(
'accessible',
'is',
'Let $\kappa$ be a regular cardinal. A category is <i>$\kappa$-accessible</i> if it has $\kappa$-filtered colimits and there is a (small) set $G$ of $\kappa$-presentable objects such that every object is a $\kappa$-filtered colimit of objects in $G$. A category is <i>accessible</i> if it is $\kappa$-accessible for some regular cardinal $\kappa$.',
'https://ncatlab.org/nlab/show/accessible+category',
'coaccessible',
TRUE
),
(
'coaccessible',
'is',
'A category is <i>coaccessible</i> if its opposite category is accessible.',
NULL,
'accessible',
TRUE
),
(
'finitely accessible',
'is',
'A category is <i>finitely accessible</i> if it has filtered colimits and there is a (small) set $G$ of finitely presentable objects such that every object is a filtered colimit of objects in $S$.',
'https://ncatlab.org/nlab/show/accessible+category',
NULL,
TRUE
),
(
'ℵ₁-accessible',
'is',
'This is the special case of the notion of $\kappa$-accessible categories, where $\kappa = \aleph_1$ is the first uncountable cardinal.',
'https://ncatlab.org/nlab/show/accessible+category',
NULL,
TRUE
),
(
'generalized variety',
'is a',
'A category is a <i>generalized variety</i> if it has sifted colimits and there is a (small) set $G$ of strongly finitely presentable objects such that every object is a sifted colimit of objects from $G$. Generalized varieties are like locally strongly finitely presentable categories but without colimits. The relation is similar as between finitely accessible and locally finitely presentable categories. This notion is defined in <a href="http://www.tac.mta.ca/tac/volumes/8/n3/8-03abs.html" target="_blank">[AR01, Def. 3.6]</a>.',
NULL,
NULL,
TRUE
),
(
'multi-algebraic',
'is',
'A category is <i>multi-algebraic</i> if it satisfies one of the following equivalent conditions:
<ol>
<li>It is a multi-cocomplete generalized variety, that is, it has multi-colimits and sifted colimits of all small diagrams, and there is a (small) set $G$ of strongly finitely presentable objects such that every object is a sifted colimit of objects from $G$.</li>
<li>It is equivalent to the category of models of a small (finite product, coproduct)-sketch, shortly small <i>FPC-sketch</i>.</li>
<li>It is equivalent to the category of multi-finite-product-preserving functors to $\mathbf{Set}$ from a small category with multi-finite-products (<i>multi-algebraic theory</i>). Here, multi-finite-products means multi-limits of finite discrete diagrams.</li>
<li>It is equivalent to the category of models of a small multi-finite-product sketch.</li>
</ol>
Multi-algebraic categories are like locally strongly finitely presentable categories but only with multi-colimits. The relation is similar as between locally finitely multi-presentable and locally finitely presentable categories.
For equivalence of conditions above, see [<a href="https://doi.org/10.1016/S0022-4049(01)00015-9" target="_blank">AR01a</a>, Lem. 1] and [<a href="http://www.tac.mta.ca/tac/volumes/8/n3/8-03abs.html" target="_blank">AR01b</a>, Thm. 4.4].
This notion was originally introduced by <a href="https://doi.org/10.1007/BF01224953" target="_blank">Diers</a>.',
NULL,
NULL,
TRUE
),
(
'locally multi-presentable',
'is',
'Let $\kappa$ be a regular cardinal. A category is <i>locally $\kappa$-multi-presentable</i> if it satisfies one of the following equivalent conditions:
<ol>
<li>It is $\kappa$-accessible and multi-cocomplete.</li>
<li>It is $\kappa$-accessible and has connected limits.</li>
<li>It is equivalent to the category of models of a small ($\kappa$-limit, coproduct)-sketch.</li>
</ol>
For equivalence of conditions above, see Thm. 4.30, Thm. 4.32, and the remark below in <a href="https://ncatlab.org/nlab/show/Locally+Presentable+and+Accessible+Categories" target="_blank">Adamek-Rosicky</a>. A category is <i>locally multi-presentable</i> if it is locally $\kappa$-multi-presentable for some $\kappa$.',
'https://ncatlab.org/nlab/show/locally+multipresentable+category',
NULL,
TRUE
),
(
'locally finitely multi-presentable',
'is',
'A category is <i>locally finitely multi-presentable</i> if it satisfies one of the following equivalent conditions:
<ol>
<li>It is finitely accessible and multi-cocomplete.</li>
<li>It is finitely accessible and has connected limits.</li>
<li>It is equivalent to the category of models of a small (finite limit, coproduct)-sketch.</li>
</ol>
For equivalence of conditions above, see Thm. 4.30, Thm. 4.32, and the remark below in <a href="https://ncatlab.org/nlab/show/Locally+Presentable+and+Accessible+Categories" target="_blank">Adamek-Rosicky</a>.',
'https://ncatlab.org/nlab/show/locally+multipresentable+category',
NULL,
TRUE
),
(
'locally poly-presentable',
'is',
'A category is <i>locally poly-presentable</i> if it is accessible and has wide pullbacks.',
'https://ncatlab.org/nlab/show/locally+polypresentable+category',
NULL,
TRUE
);