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Lov\'asz (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C.
Operations with structures
L. Lovász · 1967
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Polyadic spaces and elementary theories
A. Joyal · 1971
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Categories of continuous functors, I
P. J. Freyd and G. M. Kelly · 1972
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Direct product in locally finite categories
L. Lovász · 1972
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Isomorphism types of objects in categories determined by numbers of morphisms
A. Pultr · 1973
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Abstract and concrete categories
J. Adámek, H. Herrlich, and G. E. Strecker · 1990
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Some inequalities in hom sets
J. Isbell · 1991
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An optimal lower bound on the number of variables for graph identification
J.-Y. Cai, M. Fürer, and N. Immerman · 1992
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Locally presentable and accessible categories
J. Adámek and J. Rosický · 1994
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Handbook of categorical algebra. 1
F. Borceux · 1994
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Fractional isomorphism of graphs
M. V. Ramana, E. R. Scheinerman, and D. Ullman · 1994
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Bounded Variable Logics and Counting — A Study in Finite Models
M. Otto · 1997
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S. Mac Lane · 1998
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P. G. Kolaitis and M. Y Vardi · 2000
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P. Blackburn, M. de Rijke, and Y. Venema · 2001
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Elements of Finite Model Theory
L. Libkin · 2004
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Finite Model Theory and Its Applications
E. Grädel, P. G. Kolaitis, L. Libkin, M. Marx, J. Spencer, M. Y. Vardi, Y. Venema, and S. Weinstein · 2007
The pebbling comonad in finite model theory
S. Abramsky, A. Dawar, and P. Wang · 2017
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Relating Structure and Power: Comonadic semantics for computational resources
S. Abramsky and N. Shah · 2018
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Comonadic semantics for guarded fragments
S. Abramsky and D. Marsden · 2020
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Counting bounded tree depth homomorphisms
M. Grohe · 2020
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The Weisfeiler-Leman algorithm: an exploration of its power
S. Kiefer · 2020
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Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs
L. Mančinska and D. E. Roberson · 2020
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Homomorphism preservation theorems
B. Rossman · 2008
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On recognizing graphs by numbers of homomorphisms
Z. Dvořák · 2010
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Factorization systems
E. Riehl
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A pebbling comonad for finite rank and variable logic, and an application to the equirank-variable homomorphism preservation theorem
T. Paine · 2020
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Relating Structure and Power: Extended version
S. Abramsky and N. Shah · 2021
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Game comonads & generalised quantifiers
A. Ó Conghaile and A. Dawar · 2021
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