Add "accessible" and related notions #59
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@@ -10,40 +10,130 @@ VALUES | |||||
| ( | ||||||
| 'locally finitely presentable', | ||||||
| 'is', | ||||||
| 'A category is <i>locally finitely presentable</i> if it is cocomplete and there is a set $S$ of finitely presentable objects such that every object is a small filtered colimit of objects in $S$. This is the same as being locally $\aleph_0$-presentable.', | ||||||
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| '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 is cocomplete and there is a set of $\kappa$-presentable objects $S$ such that every object is a small $\kappa$-filtered colimit of objects in $S$. A category is <i>locally presentable</i> if it is locally $\kappa$-presentable for some regular cardinal $\kappa$.', | ||||||
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| '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 small sifted colimit of objects from $G$. | ||||||
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| 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 small $\kappa$-filtered colimits and there is a (small) set $G$ of $\kappa$-presentable objects such that every object is a small $\kappa$-filtered colimit of objects in $S$. A category is <i>accessible</i> if it is $\kappa$-accessible for some regular cardinal $\kappa$.', | ||||||
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| '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 small filtered colimits and there is a (small) set $G$ of finitely presentable objects such that every object is a small filtered colimit of objects in $S$.', | ||||||
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Colimits of a given type are small unless specified otherwise. Let's not write "small products", "small filtered colimits", etc. This is annoying and large colimits are very rare, in which case one can add the word "large" or "general". I will add this to Please adjust this also here in the other definitions. Otherwise it's inconsistent (and consistency is key for a database application) with other entries, such as the definition of the "exact filtered colimits" property. |
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| '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 small sifted colimits and there is a (small) set $G$ of strongly finitely presentable objects such that every object is a small 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 small sifted colimit of objects from $G$.</li> | ||||||
| <li>It is equivalent to the category of models of a small (finite product, small 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> | ||||||
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Owner
There was a problem hiding this comment. Aren't these items (3), (4) exactly the same? |
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| </ol> | ||||||
| Multi-algebraic categories are like locally strongly finitely presentable category but only with multi-colimits. The relation is similar as between locally finitely multi-presentable and locally finitely presentable categories. | ||||||
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| 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>.', | ||||||
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Owner
There was a problem hiding this comment. I am wondering if we should also write the category of fields as a motivating example? Very briefly, like "The category of fields is a typical example of a multi-algebraic category which is not algebraic". But it's very long already. Maybe we should write an nlab article instead and link to it. WDYT? Usually the property descriptions in CatDat are never that long (so far). |
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| NULL, | ||||||
| NULL, | ||||||
| TRUE | ||||||
| ), | ||||||
| ( | ||||||
| 'locally multi-presentable', | ||||||
| 'is', | ||||||
| 'Let $\kappa$ be a regular cardinal. A category is <i>locally $\kappa$-multi-presentable</i> if it is $\kappa$-accessible and has connected limits. It is known that a category is locally $\kappa$-multi-presentable if and only if it is equivalent to the category of models of a small limit-coproduct sketch; see 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 called <i>locally multi-presentable</i> if it is locally $\kappa$-multi-presentable for some $\kappa$, equivalently, it is accessible and has connected limits.', | ||||||
| '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 is finitely accessible and has connected limits.', | ||||||
| '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 | ||||||
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There was a problem hiding this comment. This property is currently not deduced for |
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| ); | ||||||
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I asked this before, but the comment has gone somehow: why are these conditions equivalent?