Thanks to all who replied to my posting either privately or on Categories. I'd like to clarify a few points. Ross's interpretation of what I meant, and which Vaughan agrees with, is not what I was trying to say. It is part of the picture but not the one I was promoting. Small categories play a different role in category theory than large categories. Small categories are used for indexing things. Large categories consist of the things we are indexing. Okay, syntactic was not the right word. Combinatorial might be better although that has finiteness overtones. Strict? Well, I'll just stick with small. Small is not the same as essentially small. As Jeff Egger pointed out, the category of finite sets is not small. It's not even clear what this category is. The ZFCists would say that for each set A we get a finite set {A} and another {{A}}, and Barwise might even wonder if these last two are distinct. But I'm digressing. Of course every small category can be considered as a large one (perhaps large isn't the right word either). Then two equivalent ones would be considered "the same". I don't think that this is how the working categorician works. Sometimes equivalent categories are "the same" and sometimes not. An equivalence relation is a category but we would lose something (everything?) if we identified it with equality in the quotient set. So I'm saying that, as a matter of "categorical hygiene", we could be a bit more explicit about what kind "equivalence" we are allowing. And without changing mathematical practice, I'm advocating that "small" should imply that it's okay to talk of equality of objects. As usual it helps put things in relief to generalize them. Consider category theory in a world based on a topos S. A small category is a category object in S. A large category like S or Group(S) is an indexed category, given by a pseudo-functor S^op -> Cat, or a fibration over S if you prefer. A small category C gives a large one by homming. But something is lost in the process. The object functor Cat -> Set is not a 2-functor so if you compose it with a pseudo-functor S^op -> Cat you get an assignment that doesn't preserve composition in any sense. So a general indexed category doesn't have a discrete category of objects. What you can do is take the groupoid of isomorphisms 2-functor Cat -> Gpd, so that a large has a groupoid of objects, thinking of a groupoid as a set with a 2-equality on it. An indexed category gotten from a small category gives an actual functor S^op -> Cat, so now you can compose with Ob : Cat -> Set so that a small category has a discrete groupoid of objects. And that's where the idea of considering categories with a specified "equality groupoid" of isomorphisms, and equivalences defined to have the isomorphisms from the corresponding groupoids. Small categories have only identities, so equivalence means isomorphism, and large categories have all isos. At the time there were no good examples of intermediate groupoids (just products of small and large categories and the like) but Hilbert spaces with unitary isos is a good one. Well this is my second posting in a week bringing my life-time total to three! Seems a good place to stop. Best wishes for the new year. May your happiness be large and your disappointments small. Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On 03/01/2010, at 6:57 PM, Vaughan Pratt wrote:

...

For these, one can't expect the kinds of universal constructions that large categories have,

Not following. FinSet is an essentially small category, what do you mean that it doesn't enjoy universal constructions? It's even a topos.

Dear Vaughan Part of what Bob Paré was arguing, I believe, was that we should be flexible (pun intended) about what "small" means. If "small" means "finite" then FinSet is not "essentially small". Also, "small" could mean "no more than one element". Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Bob Pare' made the excellent point that not only size but quality is relevant. I definitely agree with the spirit of his remarks. Bob happens to have used in passing the term 'syntactic'. For clarity, the use of that term needs to be sharpened to avoid misunderstanding. Actually, the term 'syntax' refers NOT to small categories such as algebraic theories or rings, but rather to their PRESENTATION by signatures or by polynomial generators, et cetera. The process of presentation is an adjoint pair quite distinct from the semantical adjoint pair: both adjoint pairs have a category of theories or of rings in common but are otherwise quite independent. In particular, syntax is NOT the adjoint of semantics. Cratylus, Chomsky, and their 21st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication. That syntax is only remotely dependent on the structure of the content that is to be communicated). Both of the functors ?--------------> theories -------------->Large categories Syntax Semantics are needed. The domain category of the first can be chosen in various useful ways: sketches or diagrams of signatures et cetera. Happy new year! Bill On Fri 01/01/10 9:48 AM , pare@mathstat.dal.ca (Robert Pare) sent:

I would like to add a few thoughts to the "evil" discussion.

My 30+ years involvement with indexed categories have led me to the following understanding. There are two kinds of categories, small and large (surprise!). But the difference is not mainly one of size. Rather it's how well we can pin down the objects. The distinction between sets and classes is often thought of in terms of size but Russell's problem with the set of all sets was not one of size but rather of the nature of sets. Once you think you have the set of all sets, you can construct another set which you had missed. I.e. the notion is changing, slippery. There are set theories where you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.)Smallness is more a question of representability: a functor may fail to be representable because it's too big (no solution set) or, more often, because it's badly behaved (doesn't preserve products, say). Subfunctorsof representables are not usually representable.

In our work on indexed categories, Schumacher and I had tried to treat this question by considering categories equipped with a groupoid of isomorphisms, which we called *canonical*, and then consider functors defined up to canonical isomorphism. In small categories only identitieswere canonical whereas in large categories, all isomorphisms were canonical.Our ideas were a bit naive and not well developed and earned us some ridicule,so we quietly stopped talking about it. Recently, Makkai developed an extensive theory of functors defined up to isomorphisms, FOLDS, but did not consider the possibility of specifying which isomorphisms ahead of time, so small categories were not included.

When I used to teach category theory, before Dalhousie made me chuck my chalk chuck, I would tell students there were two kinds of categories inpractice. Large ones which are categories of structures, corresponding tovarious branches of mathematics we wished to study. These categories supported various universal constructions, all defined up to isomorphism.Two large categories are considered to be the same if they are equivalent.It was considered impolite to ask if two objects were equal. Then there are the small categories which are used to study the large ones. These are syntactic in nature. For these, one can't expect the kinds of universal constructions that large categories have, but now it's okay, even necessary, to consider equality between objects. I went on to say that there were then four kinds of functors. Functors between large categorieswere to be thought of as constructions of one structure from another, e.g.the group ring. Functors between small categories were interpretations ofone theory in another or reindexing or rearranging. Functors from small to large categories were models or diagrams in the large one. These kindsof functors are perhaps the most important of the four, although this maybe debatable. The fourth kind, from large to small are rarer. They can be thought of as gradings or partitions of the large category.

Well, after these ramblings, perhaps my message is lost. So here it is: Small categories -> equality of objects okay Large categories -> equality of objects not okay Small is beautiful, not evil.

Bob

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Dear categorists, The question for me is not : small or large categories, but small or large structures. The Bourbaki's time was the time of small structures (groups, monoïds, rings, spaces, etc), the "categorical time" is the time of structures of structures,i.e. large structures (groupoids, complete categories, abelian catégories, topos, etc.). The bridge beetwen the firsts and the seconds is the Yoneda lemma, that is to say the introduction of logic, thus of the only true evil category : the category of sets. All the other large categorical stuctures introduced by categorists are deduced from category Set, by abstraction, generalisations, constructions or restrictions. In fact "category theory" is an (wonderfull but) inappropriate name (here the word category is only an important and historical keyword): the reason is that a lot of data (limits, classifiant object, etc) appear as properties because of their universal properties (unicity up to isomorphism). But "category theory" is an illusion, nobody studies seriouly the categorical structures (I don't know any structure theorem on the categorical structures without additional properties). That is my starting point of reflexion on this subject. I think there is a true theory of the small categories, but it is not yet born, if it must ever exist. This theory should be, not only for the 1-categories, but also for the n and omega-categories (strict or not possibly). And, for me, the Yoneda lemma is an important tool (but only a tool). Such a theory may be eventually important for the computer science (particularly for the formal languages). My best wishes for the new year, Albert burroni [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Isn't there something "evil" in the definition of dual category itself? Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

I largely agree with Bob Pare's New Year's Day posting, and disagree with those who disagree with him, although Albert Burroni is right to generalise from categories to structures.

... there were two kinds of categories in practice. [Large and small.]

I went on to say that there were then four kinds of functors. - Functors between large categories were to be thought of as constructions of one structure from another, e.g. the group ring. - Functors between small categories were interpretations of one theory in another or reindexing or rearranging. - Functors from small to large categories were models or diagrams in the large one. These kinds of functors are perhaps the most important of the four, although this may be debatable. - The fourth kind, from large to small are rarer. They can be thought of as gradings or partitions of the large category.

Small categories -> equality of objects okay Large categories -> equality of objects not okay Small is beautiful, not evil.

I would, however, like to substitute INTERNAL for SMALL. Tradionally, a "small" category is one with a "set" of objects and morphisms. However, since set theory is the problem and not the solution to the foundations of mathematics, we can avoid this by translating "set" into "object of a topos" and more generally a lex category (one with pullbacks and a terminal object") or an arithmetic universe. (I believe that there was a development of "locally internal" instead of "locally small" categories by some French categorists in the 1970s -- Burroni again maybe?.) Having done this, I would like to rescue the word "set" from the set theorists, and use it to mean "object of a topos", lex category or arithmetic universe. Vaughan Pratt's remark that "FinSet is an essentially small category" is entirely consistent with what Bob said. FinSet is a LARGE category for which there is a SMALL category "finset" (whose set of objects is N) and a WEAK EQUIVALENCE, ie a full and faithful functor finset->FinSet that is essentially surjective. This is one of the "most important" functors, according to Bob. An internal category (and more generally internal structure) of course inherits equality from its carrier, which is by definition a set (object of a topos or lex category). A large category or external structure has no carrier, and therefore no notion of equality, as Bob said. Another way of seeing the large/small or external/internal distinction is that a small or internal structure gives a NAME to the external one, which we may conversely call the SEMANTICS (meaning) of the name. So, again, I agree with Bob in identifing semantics/syntax with large/small. You might object that this is a syntax without an alphabet of symbols. Again we need an internal notion, this time an INTERNAL LANGUAGE. Unfortunately, a lot of people have muddled up the terminology here. What *I* mean by an internal language is an internal structure of a suitable kind for investigating formal grammar. For example, its set of "words" is the internal free monoid on its set of "letters", which is why we need an arithmetic universe. Regrettably, other people have used the phrase "internal language" to mean a language that is equivalent to a structure, which I call a PROPER LANGUAGE, where I have anglicised French "propre" or Italian "proprio". Given, for example, an internal CCC, it has a proper language that is an INTERNAL SIMPLY TYPED LAMBDA CALCULUS. From this we may construct the internal CATEGORY OF CONTEXTS AND SUBSTITUTIONS, which is the internal CLASSIFYING CATEGORY for the lambda calculus, in particular having an internal stucture-preserving functor to the given internal CCC, and this is an interval equivalence. (I'm not going to go into whether it is a weak or a strong equivalence -- see Section 7.6 of "Practical Foundations" for this.) Besides the book, please see also mathoverflow.net/questions/8731/categorical-foundations-without-set- theory Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ]