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I'm completing a paper about (Mitchell's) semicats (well, not exactly, but let's say so for simplicity), and as a motivational example I'd like to mention at some point that the monic/epic morphisms of the semicat of real/complex normed spaces and (linear) compact operators between them (with the obvious source and target maps and the equally obvious composition) are exactly those that one is expected to get. So my question is:

Is there anything in the literature taking the point of view of semicats in the study of compact operators, in such a way that I can cite it (at least for the sake of comparison)?

Feel free to extend the same question to other objects of interest in functional analysis such as real/complex normed spaces and (strictly) contractive linear operators or (topological) pointed spaces and compactly supported base maps. I don't expect anything like Helemskii's Lectures and Exercises on Functional Analysis, but on the other hand I find it a little bit surprising that nobody has already tried to pursue this line of thought, and arguing that the reason for this "gap" may be due to the fact that "semicats are not really more general than cats", since "there exists a functorial way to turn them into a category", is just another instance of the principle of explosion.

Added later. [1] Loosely speaking, a semicat is a not-necessarily-unital category. For what it is worth, and to the best of my knowledge, the notion was first introduced by B. Mitchell in The dominion of Isbell, TAMS, Vol. 167 (1972), 319-331. [2] Monic and epic arrows in a semicat are defined in the very same way as monic and epic arrows in categories. [3] If necessary (though I don't think so): By a compact operator between $\mathcal K$-normed modules, where $\mathcal K = (\mathbb K, |\cdot|)$ is a normed rng (here, just a rng endowed with an absolute value), I mean a triple $f: \mathcal M_1 \to \mathcal M_2$ for which $\mathcal M_i = (\mathbb M_i, \|\cdot\|_i)$ is a normed (left) module over $\mathcal K$ and $f: \mathbb M_1 \to \mathbb M_2$ is a homomorphism of (left) $\mathbb K$-modules such that the image of any bounded subset of $\mathcal M_1$ under $f$ is relatively compact in $\mathcal M_2$.

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For the future: Please always use the tags with two letter prefix if (and only if) one exists instead of creating an essentially identical tag. Thanks in advance! –  quid Feb 26 '13 at 18:26
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I gather from the second example that semicats needn't have identity morphisms. Perhaps you could append the relevant definition? –  Adam Epstein Feb 26 '13 at 18:47
    
Are the epics the operators with fin-dim codomain (hence surjective)? –  Yemon Choi Feb 26 '13 at 19:28
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@Yemon. I'm probably missing something big here, but it is known, e.g., that for any separable Banach space $\mathcal X$ (over the real/complex field) there exists a compact (linear) operator $f: \mathcal X \to \mathcal X$ that is injective and has dense range; see ams.org/journals/proc/2006-134-05/S0002-9939-05-08084-6/… (Proposition 2.1). –  Salvo Tringali Feb 26 '13 at 23:03
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@Yemon. No need for feeling embarrassed: We're men, and have all the right to make honest mistakes. Btw, here is a possibly simpler construction: The linear operator $f:\ell^2(\mathbb R)\to \ell^2(\mathbb R): (x_n)_{n=1}^\infty \mapsto (x_n/n)_{n=1}^\infty$ is compact, injective and has dense range. –  Salvo Tringali Feb 26 '13 at 23:51

1 Answer 1

See:

Peter W. Michor: Banach-Semikategorien, I. Sitzungsberichte Österreichische Akademie Wiss., Abt II, 185 (1976), 181--204, MR 56#3644 a, ZM 359.46049. (pdf of I)

Peter W. Michor: Banach-Semikategorien, II. Sitzungsberichte Österreichische Akademie Wiss., Abt II, 185 (1976), 205--219, MR 56#3644 b, ZM 359.46050. (pdf of II)

Peter W. Michor: Banach-Semikategorien, III. Sitzungsberichte Österreichische Akademie Wissenschaften., Abt II, 185 (1976), 221--238, MR 56#3644 c, ZM 359.46051. (pdf of III)

Johann Cigler, Viktor Losert, Peter W. Michor: Banach modules and functors on categories of Banach spaces. Lecture Notes in Pure and Applied Mathematics 46, Marcel Dekker Inc., New York, Basel, (1979), MR 80j:46112, Zbl 411.46044. Review in Bull. AMS 3,2 (1980) (pdf)

Peter W. Michor: Functors and categories of Banach spaces. Springer Lecture Notes 651, (1978), vi+99 pp., MR 80h:46116, Zbl 369.46069. (pdf)

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Thank you for the references! I'm very interested, but unluckily for me, I cannot read German. So let me ask: Is there any hope that the material of the three papers of the series Banach-Semikategorien has been integrally included in the books mentioned in your answer? In other words, is there anything relevant in the papers that cannot be found in the books? –  Salvo Tringali Mar 8 '13 at 17:03
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Sorry, nothing from the 3 papers is included in the books. The main thrust of the three papers are Banach semicategories with approximate units. The MathSciNet review was quite detailed, as I recall. –  Peter Michor Mar 8 '13 at 17:18
    
Thanks, I will try to understand as much as I can (hopefully with the help of my German office mate). –  Salvo Tringali Mar 8 '13 at 17:51

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