What are the limits in the span categories? and what is known about them in the literature?
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1$\begingroup$ Some quick basics are: in as far as they exist limits are also colimits, and vice versa. If the underlying category $\mathcal{C}$ is extensive then the coproduct/product in $Corr(\mathcal{C})$ is given by disjoint union in $\mathcal{C}$. ncatlab.org/nlab/show/span#LimitsAndColimits $\endgroup$– Urs SchreiberCommented Aug 12, 2013 at 6:39
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$\begingroup$ these are bicategories. A. Carboni, S. Kasangian and R. Street, Bicategories of spans and relations, J. Pure and Appl. Algebra, 33 (1984) B. J. Day, Limit Spaces and Closed Span Categories, Lecture Notes in Math.,420 (Springer, Berlin-New York, 1974), 65-74. $\endgroup$– Buschi SergioCommented Aug 14, 2013 at 11:57
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I can think of two interesting classes of limits in bicategories of spans (which, as Urs mentioned, are necessarily also colimits, since $\mathrm{Span}(C)$ is equivalent to its opposite).
The first is a generalization of Urs's comment: any van Kampen colimit in $C$ is a (co)limit in $\mathrm{Span}(C)$ — and conversely, this property characterizes van Kampen colimits. This is in
- Pawel Sobocinski and Tobias Heindel, Being Van Kampen is a universal property, arXiv:1101.4594
The second is that $\mathrm{Span}(C)$ always has Eilenberg-Moore objects for comonads (a sort of 2-categorical limit) — and this figures in a characterization of bicategories of the form $\mathrm{Span}(C)$. This is in
- Stephen Lack, R.F.C. Walters, and R.J. Wood, Bicategories of spans as cartesian bicategories, TAC 24 (1).
answered Aug 13, 2013 at 23:33
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$\begingroup$ Dear Prof. Mike Shulman, I write some notes about Gray-tensor of 2-categories and pseudofunctor (time ago you ask her about a generalization of Gray product to pseudofunctor), My EM is "[email protected]" mine is a little work, but if you want I traslate (as I can) and send you. $\endgroup$ Commented Aug 14, 2013 at 12:29
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$\begingroup$ That's off-topic here, so please email me directly or post a comment at the question in question. $\endgroup$ Commented Aug 15, 2013 at 16:18