#
Multilinearity of Sketches

##
David B. Benson

We give a precise characterization for when
the models of the tensor product of sketches are
structurally isomorphic to
the models of either sketch in the models of the other.
For each base category K
call the just mentioned property (sketch) K-multilinearity.
Say that two sketches are K-compatible with respect to base
category K just in case
in each K-model, the limits for each limit specification
in each sketch commute with
the colimits for each colimit specification in the other sketch
and all limits and colimits are pointwise.
Two sketches are K-multilinear if and only
if the two sketches are K-compatible.
This property then extends to strong Colimits of sketches.

We shall use the technically useful property of limited
completeness and completeness of every category of models
of sketches. That is, categories of sketch models have
all limits commuting with the sketched colimits and
and all colimits commuting with the sketched limits.
Often used implicitly, the precise statement
of this property and its proof appears here.

Keywords: categorical model theory, Ehresmann sketches, data structures.

1991 MSC: 18C10, 68Q65, 03C52, 18A25, 68P05.

*Theory and Applications of Categories*, Vol. 3, 1997, No. 11, pp 267-277.

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